A Unifying Construction for Difference Sets James A

University of Richmond UR Scholarship Repository

Math and Computer Science Faculty Publications Math and Computer Science

10-1997 A Unifying Construction for Difference Sets James A. Davis University of Richmond, [email protected]

Jonathan Jedwab

Follow this and additional works at: http://scholarship.richmond.edu/mathcs-faculty-publications Part of the Discrete Mathematics and Combinatorics Commons

Recommended Citation Davis, James A., and Jonathan Jedwab. "A Unifying Construction for Difference Sets." Journal of Combinatorial Theory, Series A 80, no. 1 (October 1997): 13-78. doi: 10.1006/jcta.1997.2796.

This Article is brought to you for free and open access by the Math and Computer Science at UR Scholarship Repository. It has been accepted for inclusion in Math and Computer Science Faculty Publications by an authorized administrator of UR Scholarship Repository. For more information, please contact [email protected]. Journal of Combinatorial Theory, Series A TA2796 journal of combinatorial theory, Series A 80, 1378 (1997) article no. TA972796

A Unifying Construction for Difference Sets

James A. Davis*

Department of Mathematics and Computer Science, University of Richmond, Virginia 23173

and

Jonathan Jedwab

HewlettPackard Laboratories, Filton Road, Stoke Gifford, Bristol BS12 6QZ, United Kingdom

Communicated by the Managing Editors

Received July 29, 1996

We present a recursive construction for difference sets which unifies the Hadamard, McFarland, and Spence parameter families and deals with all abelian groups known to contain such difference sets. The construction yields a new family of difference sets with parameters (v, k, *, n)=(22d+4(22d+2&1)Â3, 22d+1(22d+3+1)Â3, 22d+1(22d+1+1)Â3, 24d+2) for d0. The construction establishes that a McFarland difference set exists in an abelian group of order 22d+3(22d+1+1)Â3 if and only if the Sylow 2-subgroup has exponent at most 4. The results depend on a second recursive construction, for semi-regular relative difference sets with an elementary abelian forbidden subgroup of order pr. This second construction deals with all abelian groups known to contain such relative difference sets and significantly improves on previous results, particularly for r>1. We show that the group order need not be a prime power when the forbidden subgroup has order 2. We also show that the group order can grow without bound while its Sylow p-subgroup has fixed rank and that this rank can be as small as 2r. Both of the recursive constructions generalise to nonabelian groups. 1997 Academic Press

1. INTRODUCTION

A k-element subset D of a finite multiplicative group G of order v is called a (v, k, *, n)-difference set in G provided that the multiset of

* The author thanks HewlettPackard for their generous hospitality and support during his sabbatical year 19951996. This work is also partially supported by NSA Grant MDA904- 94-2062. 13 0097-3165Â97 25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved.

File: 582A 279601 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3880 Signs: 2052 . Length: 50 pic 3 pts, 212 mm 14 DAVIS AND JEDWAB

&1 ``differences'' [d1d2 | d1 , d2 # D, d1{d2] contains each nonidentity element of G exactly * times; we write n=k&*. For example, D=[x, x2, x4] 7 is a (7, 3, 1, 2)-difference set in Z7=(x | x =1). It is easy to check that [y, x, xy, xy2, x2y, x3y3], [z, xz, z2, yz2, z3, xyz3] and [z, xz, w, yw, zw, xyzw] are examples of (16, 6, 2, 4)-difference sets in the abelian groups 2 4 4 2 4 2 2 4 Z4=(x, y | x =y =1), Z4_Z2=(x, y, z | z =x =y =1) and Z2= (x, y, z, w | x2=y2=z2=w2=1) respectively. Difference sets arise in a wide variety of theoretical and applied contexts. They are important in design theory because a (v, k, *, n)-difference set in G is equivalent to a symmetric (v, k, *, n)-design with a regular automorphism group G [32]. The study of difference sets is also deeply connected with coding theory because the code, over a field F, of the symmetric design corresponding to a (v, k, *, n)-difference set may be considered as the right ideal generated by D in the group algebra FG [29, 32]. Difference sets in abelian groups are the natural solution to many problems of signal design in digital communications, including synchronisation [25], radar [1], coded aperture imaging [23, 52], and optical image alignment [41]. For a recent survey of difference sets see Jungnickel [29]. The central problem is to determine, for each parameter set (v, k, *, n), which groups of order v contain a difference set with these parameters. An extensive literature has been devoted to this problem, exposing con- siderable interplay between difference sets and such diverse branches of mathematics as algebraic number theory, character theory, representation theory, finite geometry and graph theory. Nonetheless the central problem remains open, both for abelian and nonabelian groups, except for heavily restricted parameter sets. One of the most popular techniques for constructing a difference set or for ruling out its existence is to consider the image of a hypothetical difference set under mappings from the group G to one or more quotient groups GÂU (see Ma and Schmidt [40] for a recent example). By a counting argument the parameters (v, k, *, n) of a difference set are related by k(k&1)=*(v&1). We can assume that kvÂ2 because D is a (v, k, *, n)-difference set in G if and only if the complement G"D is a (v, v&k, v&2k+*, n)-difference set in G. The trivial cases k=0 and k=1 are usually excluded (although we shall use trivial examples as the initial case of some recursive constructions). Besides these constraints, difference sets are classified into families according to further relationships between the parameters. A great deal of research on difference sets has focussed on two particular families of parameters: the Hadamard family given by

(v, k, *, n)=(4N2, N(2N&1), N(N&1), N2)(1)

File: 582A 279602 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3258 Signs: 2777 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 15 for integer N1 (see Davis and Jedwab [13] for a survey), and the McFarland family given by qd+1&1 qd+1&1 qd&1 (v, k, *, n)= qd+1 +1 , qd , qd , q2d (2) \ \ q&1 + \ q&1 + \q&1 + + for q a prime power and integer d0 (see Ma and Schmidt [39] for a sum- mary and new results). The Hadamard family derives its name from the fact that D is a Hadamard difference set if and only if the (+1, &1) incidence matrix of the design corresponding to D is a regular Hadamard matrix [29, 55]. The Hadamard and McFarland families intersect in 2-groups: the Hadamard family with N=2d corresponds to the McFarland family with q=2. The most recent discovery of a new family of parameters for which difference sets exist was given by Spence [54] in 1977:

3d+1&1 3d+1+1 3d+1 (v, k, *, n)= 3d+1 ,3d ,3d ,32d (3) \ \ 2 + \ 2 + \ 2 + + for integer d0. Other families of difference set parameters include the projective geometries, the PaleyHadamard family, and the twin prime power family [4]. For each of these parameter families, difference sets have been constructed for infinitely many values of the parameters, but not necessarily in all possible groups of each order. A notable exception is abelian 2-groups, for which Kraemer [31] completely solved the central problem: a Hadamard difference set exists in an abelian group G of order 22d+2 if and only if d+2 exp(G)2 . (The exponent of a group G with identity 1G , written : exp(G), is the smallest integer : for which g =1G for all g # G.) A powerful stimulus to the discovery of new results on difference sets has been the identification of open cases in groups of relatively small order. For example, Dillon [18] led a research programme to examine constructions and nonexistence results for Hadamard difference sets in all 267 groups of order 64, which highlighted a single outstanding case. The solution of this last case by Liebler and Smith [35] demonstrated that Turyn's exponent bound [55] of 2d+2 for Hadamard difference sets in abelian groups of order 22d+2 can be exceeded in the nonabelian case. A subsequent collaborative effort to examine Hadamard difference sets in groups of order 100 led to Smith's surprising discovery [53] of a nonabelian group of order 100 which contains a Hadamard difference set even though no abelian group of this order does so! Another example is the table of existence of difference sets in abelian groups with k50 produced by Lander [32], of which the last open cases have recently been settled by J. E. Iiams [private communication, 1994]. In 1992 Jungnickel [29] modified the parameter range of this table to n30 and listed three open cases, the last of which

File: 582A 279603 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3228 Signs: 2626 . Length: 45 pic 0 pts, 190 mm 16 DAVIS AND JEDWAB was settled when Arasu and Sehgal [3] exhibited a (96, 20, 4, 16) 2 McFarland difference set in Z4_Z2_Z3 , having q=4 and d=1. This was the first example of a McFarland difference set having q=2r>2 in a group (d+1)r+1 whose Sylow 2-subgroup does not have the form Z2 , as constructed (d+1)r&1 by McFarland [42], or Z4_Z2 , as constructed by Dillon [20]. Arasu and Sehgal's discovery sparked a search for a similar McFarland difference set in a larger group, for which the most likely candidate was 2 3 generally presumed to be a (640, 72, 8, 64)-difference set in Z4_Z2_Z5 or 3 Z4_Z2_Z5 , having q=8 and d=1. We were unable to settle these cases but managed to construct new difference sets by transferring partial results from one of these groups to a group of order 320. This new example led to many insights and eventually to the results reported in this paper. In retrospect we believe that the natural generalisation of Arasu and Sehgal's example has q=4 and d>1 rather than q>4 and d=1. In this paper we present a recursive construction for difference sets which, for the first time, unifies the Hadamard, McFarland, and Spence families. No abelian group known to contain a difference set with parameters from one of these families lies outside the scope of this result (although certain initial examples required for the Hadamard family must be constructed separately). The construction also yields the new parameter family 22d+2&1 22d+3+1 (v, k, *, n)= 22d+4 ,22d+1 , \ \ 3 + \ 3 + 22d+1+1 22d+1 ,24d+2 (4) \ 3 + + for integer d0, for which the smallest previously unknown abelian 6 4 2 2 examples occur in the groups Z2_Z5 , Z4_Z2_Z5 and Z4_Z2_Z5 with parameters (320, 88, 24, 64). This family of difference sets also represents a new family of symmetric designs with the same parameters (4). In addi- tion, the construction establishes that a McFarland difference set with q=4 exists in an abelian group of order 22d+3(22d+1+1)Â3 if and only if the Sylow 2-subgroup has exponent at most 4. This necessary and sufficient condition is analogous to Kraemer's [31] result for the case q=2. The 2 3 smallest previously unknown examples occur in the groups Z4_Z2_Z11 3 and Z4_Z2_Z11 with parameters (1408, 336, 80, 256). The essential idea of the construction is to combine multiple copies of a difference set with a semi-regular relative difference set to generate a difference set in a larger group. A preliminary announcement of these results was given in [12]. A k-element subset R of a finite multiplicative group G of order mu containing a normal subgroup U of order u is called a (m, u, k, *) relative difference set (RDS) in G relative to U provided that the multiset

File: 582A 279604 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3288 Signs: 2659 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 17

&1 [r1r2 | r1 , r2 # R, r1{r2] contains each element of G"U exactly * times and contains no element of U. The subgroup U is sometimes called the forbidden subgroup. (We have not used the conventional notation N for the normal subgroup and n for its order to avoid confusion with the difference set parameter n.) For example, R=[1, x, y, xy3, z, xy2z, x2y3z, x3y3z] is 2 4 4 2 a (8, 4, 8, 2) RDS in Z4_Z2=(x, y, z|x =y =z =1) relative to 2 2 2 (x , y )$Z2.A(m,u,k,*) RDS in G, relative to some normal subgroup U, is equivalent to a square divisible (m, u, k, *)-design whose automorphism group G acts regularly on points and blocks [28]. For a recent survey of RDSs see Pott [48]. The central problem is to determine, for each parameter set (m, u, k, *), the groups G of order mu and the normal subgroups U of order u for which G contains a RDS relative to U with these parameters. By a counting argument the parameters (m, u, k, *) of a RDS are related by k(k&1)=u*(m&1). If k=u* then the RDS is called semi-regular and the parameters are (u*, u, u*, *). In contrast to the situation for difference sets, the complement G"R of a RDS R is not in general a RDS. The trivial cases k=0 and k=1 are usually excluded. A difference set can be considered as a RDS with u=1. Furthermore, the image of a (m, u, k, *) RDS in G relative to U under the quotient mapping from G to GÂU is a (m, k, u*, k&u*)-difference set in GÂU. In particular, the image of a semi-regular (u*, u, u*, *) RDS in G relative to U is a trivial (u*, u*, u*, 0)-difference set in GÂU. A great deal of attention has been paid to semi-regular RDSs in p-groups, whose parameters have the form ( pw, pr, pw, pw&r) for p prime. Indeed, Pott [48] calls the central problem for these RDSs ``one of the most interesting questions about RDSs.'' Ma and Schmidt [37] have recently solved the central problem for r=1 in the abelian case, with some exceptions when w and p are odd, but describe the case r>1 as ``much more difficult.'' In this paper we present a recursive construction for semi-regular RDSs r in a group G relative to a subgroup U$Z p, where the Sylow p-subgroup of G has rank at least 2r. The essential idea is to combine semi-regular RDSs in pr quotient groups of G to form a semi-regular RDS in G. This RDS construction produces the families of RDSs needed for the recursive difference set construction. It also establishes an extensive pattern of exist- r ence for semi-regular RDSs relative to subgroups U$Zp, for each r1. This significantly improves on the previous state of knowledge for semi- regular RDSs, particularly when r>1. The position of the subgroup U within the group G is of crucial importance in these results. We show that the order of G can grow without bound while its Sylow p-subgroup has fixed rank and that this rank can be as small as 2r. In the case pr=2 we obtain results for groups G whose order need not be a prime power, so that the RDS parameters have the form (2*,2,2*,*) where * need not be a

File: 582A 279605 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3449 Signs: 3007 . Length: 45 pic 0 pts, 190 mm 18 DAVIS AND JEDWAB power of 2. In all other cases our results are for p-groups G and so the RDS parameters have the form ( pw, pr, pw, pw&r) for p prime. In particular, we improve on Ma and Schmidt's result [37] to show that there exists a ( pw, p, pw, pw&1) semi-regular RDS in any abelian group G of order pw+1 relative to any subgroup U of order p except possibly when p is odd, 2 w=2d+1 is odd, and either G=Zp d+1 or G=U_Zp d+1_Zpd. We are not aware of any abelian groups G known to contain semi-regular RDSs relative to an elementary abelian subgroup which are not covered by our results. Difference sets are usually studied in the context of the group ring Z[G] of the group G over the ring of integers Z. The definition of a (v, k, *, n)- (&1) difference set D in G is equivalent to the equation DD =n1G+*G in Z[G], where by an abuse of notation we have identified the sets D, D(&1), (&1) &1 G with the respective group ring elements D=d # D d, D =d # D d , G=g # G g, and 1G is the identity of G. Similarly the definition of a (m, u, k, *) RDS R in G relative to U is equivalent to the equation (&1) RR =k1G+*(G&U)inZ[G]. An alternative viewpoint for considering difference sets and RDSs, predominant in engineering papers, is via the correlation properties of binary arrays [27]. The (1, 0) binary array A corresponding to a subset

D of G is (ag | g # G) defined by ag=1 if g # D and ag=0 if g Â D. (&1) Then DD =g # G RA(g)g in Z[G], where RA(g)=h # G ahagh is the autocorrelation of the array A at displacement g. The (+1, &1) binary array B=(bg | g #G) is given by the linear transformation bg=1&2ag. When G is abelian the binary arrays A and B can be represented as matrices. Although binary arrays do not appear in this exposition we have found the (+1, &1) matrix representation to be an invaluable tool for visualisation. We now give some definitions and results which will be used freely throughout the paper without further reference. We shall follow the prac- tice (standard in the difference set literature) of abusing notation by iden- tifying sets with group ring elements, as described above. Since we shall be concerned principally with abelian groups, all groups will be implicitly abelian unless otherwise stated. We write >r Z for the direct product i=1 :i Z _Z _}}}_Z . For w a positive integer and p prime, we call p self- :1 :2 :r i conjugate modulo w if p #&1 (mod wp) for some integer i, where wp is the largest divisor of w coprime to p. In the abelian case, a character of the group G is a homomorphism from G to the multiplicative group of com- plex roots of unity. Under pointwise multiplication the set G* of characters of G forms a group isomorphic to G. The identity of this group is the principal character that maps every element of G to 1. The character sum of a character / over the group ring element C corresponding to a subset of G is /(C)=c # C /(c). It is well-known that the character sum /(C) is 0 for

File: 582A 279606 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3426 Signs: 2898 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 19 all nonprincipal characters / of G if and only if C is a multiple of G

(regarded as a group ring element), and that / # G* /(g) is nonzero if and only if g=1G. If a character / is nonprincipal on G and principal on a subgroup U then / induces a nonprincipal character on GÂU defined by

(gU)=/(g). ( is well-defined because if g1U=g2U then g1=ug2 for some u # U and /(u)=1 for every element u of U.) The use of character sums to study difference sets in abelian groups was introduced by Turyn in his seminal paper [55] and subsequently extended to RDSs:

Lemma 1.1. (i) The k-element subset D of an abelian group G of order visa(v,k,*,n)-difference set in G if and only if |/(D)|=-n for every nonprincipal character / of G. (ii) The k-element subset R of an abelian group G of order mu containing a subgroup U of order u is a (m, u, k, *) RDS in G relative to U if and only if for every nonprincipal character / of G

-k if / nonprincipal on U |/(R)|= {-k&u* if / principal on U.

The existence of a subset D or R in Lemma 1.1 with the character properties described forces the implicitly defined parameters n and * respectively to be integer. Lemma 1.1 indicates a general strategy for constructing difference sets and RDSs, namely to choose a group subset for which all nonprincipal character sums have the correct modulus. In Section 2 we show that the determination of character sums can be greatly facilitated by selecting the group subset as a collection of ``building blocks'' which inter- act in a simple way. This formalises many ideas which have been used implicitly in previous papers. At the end of Section 2 we give an overview of the paper in terms of the concepts introduced.

2. BUILDING SETS

Many of the key ideas in this paper were developed from studying a construction due to McFarland [42], and a modification given by Dillon [20], for difference sets with parameters (2). The construction regards the elementary abelian group G of order qd+1 as a vector space P of dimension d+1 over GF(q), where q is a prime power. There are h=(qd+1&1)Â

(q&1) subspaces H0 , H1 , ..., Hh&1 of P of dimension d, called hyperplanes. Let G$ be any group (not necessarily abelian) containing G as a central

File: 582A 279607 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 2842 Signs: 2215 . Length: 45 pic 0 pts, 190 mm 20 DAVIS AND JEDWAB

subgroup of index h+1 and let g$0 , g$1 , ..., g$h # G$ be coset representatives h&1 of G in G$. Then i=0 g$i Hi is a McFarland difference set in G$. The construction can be viewed as depending crucially on the following property: for any nonprincipal character of G there is exactly one hyperplane Hi having a nonzero character sum, and this nonzero character sum always has the same modulus qd. The difference set is comprised of h+1 subsets of G, namely the hyperplanes together with the empty set. In the case when G$ is abelian, Lemma 1.1(i) can be used to verify that the construction produces a difference set, as follows. For characters of G$ which are nonprincipal on G, the required character sum modulus of - n=qd is provided by a contribution of qd from one subset and 0 from all the other subsets. For nonprincipal characters of G$ which are principal on G, we shall see that the required character sum modulus of qd follows easily as a consequence of the subset sizes. A construction for semi-regular RDSs depending on the same property, in the case d=1, is due to Davis [10]. Let G$ be any group (not necessarily abelian) containing G as a central subgroup of index h&1=q h&1 and let g$1 , g$2 , ..., g$h&1 be coset representatives of G in G$. Then i=1 g$i Hi 2 2 is a (q , q, q , q) semi-regular RDS in G$ relative to H0 . The RDS is comprised of h&1 subsets of G, namely h&1 of the h hyperplanes. In the case when G$ is abelian, Lemma 1.1(ii) can be used to verify that the construction produces a RDS, as follows. If a nonprincipal character / of G is principal on H0 then each subset provides a contribution to the character sum modulus of 0, and if / is nonprincipal on H0 then one subset contributes - k=q and the rest contribute 0. This gives the required character sum modulus for characters of G$ which are nonprincipal on G. For nonprincipal characters of G$ which are principal on G, the required character sum modulus of 0 is again a consequence of the subset sizes. [10] reports an observation of Pott's that, in the case d>1, the same construction produces a (q(h&1), qd, qd(h&1), qd+1((qd&1&1)Â(q&1)), qd((qd&1)Â

(q&1))) semi-regular divisible difference set in G$ relative to H0 because of the mutual character properties of the hyperplanes (see Jungnickel [28] for a definition and discussion of divisible difference sets). Motivated by these examples, we define a building block in a group G with modulus m to be a subset of G such that all nonprincipal character sums over the subset have modulus either 0 or m. Here and subsequently in the paper, all groups will be implicitly assumed to be abelian unless otherwise stated. Some examples of building blocks are a coset of a subgroup of G, a semi-regular RDS in G relative to a subgroup U, and a difference set in G. For integers a1 and t1 we define a (a, m, t) building set (BS) on a group G relative to a subgroup U to be a collection of t building blocks in G with modulus m, each containing a elements, such that for every nonprincipal character / of G

File: 582A 279608 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3550 Signs: 3057 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 21

(i) exactly one building block has nonzero character sum if / is nonprincipal on U and (ii) no building block has nonzero character sum if / is principal on U.

It follows immediately from Lemma 1.1(ii) and the relationship between RDS parameters that, for a>1, a (a, -a, 1) BS on a group G relative to a subgroup U of order u is equivalent to a (a, u, a, aÂu) semi-regular RDS in G relative to U. (A trivial (1, 1, 1) BS is equivalent to a (1, u, 1, 0) RDS.) We now show that a BS on a group G relative to a subgroup U can be used to construct a BS on larger groups containing G as a subgroup. In particular we shall construct a semi-regular RDS as a single building block on a group containing G.

Lemma 2.1. Suppose there exists a (a, - at, t) BS on a group G relative to a subgroup U. Then there exists a (as, - at, tÂs) BS on G$ relative to U, where s divides t and G$ is any group containing G as a subgroup of index s.

Proof. Let [B1, B2, ..., Bt] be a (a, - at, t)BSonGrelative to U. For s each j=1, 2, ..., tÂs define the subset Rj=i=1 g$i Bi+( j&1)s of G$, where g$1 , g$2 , ..., g$s # G$ are coset representatives of G in G$. Let / be a nonprincipal character of G$ and consider the character sum /(Rj)= s i=1 /(g$i) /(Bi+(j&1)s). We distinguish three cases: / is principal on G and nonprincipal on G$; / is principal on U and nonprincipal on G; and / is nonprincipal on U. In the first case, when / is principal on G and nonprincipal on G$(sos>1), /(Bi+( j&1)s)=|Bi+( j&1)s |=a for each ordered pair s (i, j ) and so /(Rj )=ai=1 /(g$i )=0 for each j. The last equality uses the fact that / induces a nonprincipal character on G$ÂG, and the associated character sum over this group is 0. In the second case, when / is principal on U and nonprincipal on G, by assumption /(Bi+( j&1)s)=0 for each ordered pair (i, j ) and so again /(Rj)=0 for each j. In the third case, when

/ is nonprincipal on U, by assumption |/(Bi+( j&1)s)| equals - at for exactly one ordered pair (i, j ) (say (I, J)) and equals 0 for all other ordered pairs (i, j). Therefore |/(RJ)|=|/(g$I)| |/(BI+(J&1)s)|=-at and |/(Rj )|=0 for each j{J.

The character sums for the three cases show that [R1, R2, ..., RtÂs] is a (as, - at, tÂs)BSonG$ relative to U. K

Note that, in the proof of Lemma 2.1, the building blocks Bi can have non-empty intersection but by definition no set Rj contains repeated elements. We next show that in the case s=t of Lemma 2.1 we can obtain a semi-regular RDS in G$ from a BS on G, and we shall exploit this result in Section 8 to deduce the existence of semi-regular RDSs from BSs.

File: 582A 279609 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3312 Signs: 2561 . Length: 45 pic 0 pts, 190 mm 22 DAVIS AND JEDWAB

Theorem 2.2. Suppose there exists a (a, - at, t) BS on a group G relative to a subgroup U of order u, where at>1. Then there exists a (at, u, at, atÂu) semi-regular RDS in G$ relative to U, where G$ is any group containing G as a subgroup of index t.

Proof. Apply Lemma 2.1 with s=t to obtain a (at, - at,1)BSonG$ relative to U. For at>1, this is equivalent to a (at, u, at, atÂu) semi-regular RDS in G$ relative to U. K

For example, the hyperplane construction of Davis [10] and Pott (reported in [10]) discussed at the beginning of this section can be inter- d d preted as a (q , q , h&1) BS [H1, H2, ..., Hh&1] on G relative to H0 , where G is the elementary abelian subgroup of order qd+1 and h= (qd+1&1)Â(q&1). By Theorem 2.2, the case d=1 implies the existence of a(q2,q,q2,q) semi-regular RDS in any group G$ containing G as a subgroup of index q. We shall develop a powerful generalisation of this hyperplane construction in Section 4. In this paper we consider only the case m=- at of a (a, m, t) BS. We have given the more general definition because of the apparent connection with divisible difference sets. It seems that many other known constructions for divisible difference sets can be analysed in terms of BSs. A second example, due to Arasu and Sehgal [3], is a (8, 4, 2) BS on 2 4 4 2 2 2 2 Z4_Z2=(x, y, z|x =y =z =1) relative to (x , y )$Z2, where the building blocks are [1, x, xz, x2z, x2yz, xy, xy3z, y3] and [1, x3, x3y2z, x2y2z, yz, xy3z, xy3, x2y]. By Theorem 2.2, this implies the existence of a 3 (16, 4, 16, 4) semi-regular RDS in each of the groups Z8_Z4_Z2 , Z4, and 2 2 2 Z4_Z2 relative to a subgroup isomorphic to Z2 contained within two of the largest direct factors of the group.

Given a (a, - at, t)BS[B1, B2, ..., Bt] on a group G relative to a subgroup U of order u we can find several constraints on the parameters a, t, u and |G|. Theorem 2.2 implies that u | at for at>1 (which derives from the condition that * be integer in Lemma 1.1(ii)). Theorem 2.2, together with the trivial case at=1, also implies that |G|=ua. Furthermore we can show that t | a(u&1), as we now outline. Let G* be the group of characters of G. For any building block B , |/(B )|2= /(g ) /(g )= i / # G* i / # G* g1 #Bi g2 # Bi 1 2 /(g g&1)= /(1 )=|B |}|G|. Therefore if / #G* g1 #Bi g2 #Bi 1 2 g1 # Bi / # G* G i wi is the number of nonprincipal characters of G giving a nonzero character 2 2 sum over Bi then a +atwi=ua , so that wi=a(u&1)Ât for all i. We have seen how to construct a semi-regular RDS from a BS. We now define a modification of a BS for the purpose of constructing difference sets in an analogous way. For integers a0, m1, and h1, we define a (a, m, h,+)extended building set (EBS) on a group G with respect to a subgroup U to be a collection of h building blocks in G with modulus m,of

File: 582A 279610 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3577 Signs: 2767 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 23 which h&1 contain a elements and one contains a+m elements, such that for every nonprincipal character / of G (i) exactly one building block has nonzero character sum if / is principal on U and (ii) no building block has nonzero character sum if / is nonprincipal on U. We define a (a, m, h, &) EBS on G with respect to U in the same way, with a+m replaced by a&m. We shall treat both cases simultaneously by referring to a (a, m, h, \) EBS. Notice that the role of principal and nonprincipal characters on U is the reverse of that used in the definition of a BS. Notice also that for a EBS we must have m integer, because one building block contains a\m elements, whereas for a BS m need not be integer. We call the EBS covering in the case U=[1G], when exactly one building block has nonzero character sum for every nonprincipal character of G. (The use of ``covering'' refers not to the intersection or union of the building blocks but to their character properties.) It follows immediately from Lemma 1.1(i) that a (a, m, 1, \) covering EBS on a group G is equivalent to a (|G|, a\m, a\m&m2, m2)-difference set in G. We now show that a covering EBS on a group G can be used to construct a covering EBS on larger groups containing G as a subgroup. In particular we shall construct a difference set as a single building block on a group containing G. We shall exploit this result in Section 5 to deduce the existence of difference sets from covering EBSs.

Lemma 2.3. Suppose there exists a (a, m, h,\) covering EBS on a group G. Then there exists a (as, m, hÂs,\) covering EBS on G$, where s divides h and G$ is any group containing G as a subgroup of index s.

Proof. The proof is modelled on that of Lemma 2.1. Let [B1, B2, ..., Bh] be a (a, m, h, \) covering EBS on G and let the building block containing a\m elements be B1 . For each j=1, 2, ..., hÂs let Dj be the s subset i=1 g$i Bi+( j&1)s of G$, where g$1 , g$2 , ..., g$s # G$ are coset representatives of G in G$. If / is a principal character of G and nonprincipal s on G$(sos>1) then /(Dj)=/(g$1)|B1+( j&1)s|+i=2 /(g$i )|Bi+( j&1)s |= s /(g$1)(|B1+( j&1)s |&a)+ai=1 /(g$i)=/(g$1)(|B1+( j&1)s |&a), because the character induced by / on G$ÂG is nonprincipal. Therefore |/(Dj )| equals m for j=1 and equals 0 for all j>1. If / is a nonprincipal character of G then by assumption |/(Bi+( j&1)s)| equals m for exactly one ordered pair (i, j ) and equals 0 for all other ordered pairs (i, j ). Therefore |/(Dj )| equals m for exactly one value of j and equals 0 for all other values of j.

Combining these two cases, [D1, D2, ..., DhÂs] is a (as, m, hÂs,\) covering EBS on G$. K

File: 582A 279611 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3306 Signs: 2623 . Length: 45 pic 0 pts, 190 mm 24 DAVIS AND JEDWAB

Theorem 2.4. Suppose there exists a (a, m, h,\) covering EBS on a group G. Then there exists a (h |G|, ah\m, ah\m&m2, m2)-difference set in any group G$ containing G as a subgroup of index h. Proof. Apply Lemma 2.3 with s=h to obtain a (ah, m, 1, \) covering EBS on G$. This is equivalent to a (h |G|, ah\m, ah\m&m2, m2)- difference set in G$. K For example, the hyperplane construction of McFarland [42] and Dillon [20] discussed at the beginning of this section can be interpreted as d d a(q,q,h+1, &) covering EBS [,, H0, H1, ..., Hh&1] on the elementary abelian group G of order qd+1, where h=(qd+1&1)Â(q&1). By Theorem 2.4, this implies the existence of a McFarland difference set in any group G$ containing G as a subgroup of index h+1.

We have already given an example of a (8, 4, 2) BS [B1, B2] on 2 4 4 2 2 2 2 Z4_Z2=(x, y, z | x =y =z =1) relative to (x , y )$Z2, due to 2 2 Arasu and Sehgal [3]. If we define a third building block B3=(x , y ) then [B1, B2, B3] is a (8, 4, 3, &) covering EBS and then by Theorem 2.4 2 there exists a (96, 20, 4, 16)-difference set in Z4_Z2_Z3 . The main purpose of [3] was to demonstrate the existence of such a difference set, but the embedded (8, 4, 2) BS will be of great use in the construction of families of difference sets and semi-regular RDSs in later sections. We now derive some constraints on the parameters a, m, h and |G|of a(a,m,h, \) covering EBS on a group G. The relationship between difference set parameters (v, k, *, n) can be written as k2&n=*v, so from Theorem 2.4 we find a(ah\2m)=(ah\m&m2)|G|. Provided *=ah\m&m2>0, this implies that ah\m&m2 divides a(ah\2m). (The exceptional case *=0 corresponds to k=1, namely a trivial (0, 1, h,+) or (2, 1, 1, &) or (1, 1, 2, &) covering EBS on G.) Let the covering

EBS be [B1, B2, ..., Bh] and let B1 be the building block containing a\m elements. By the same argument as previously used for a BS, 2 / # G* |/(Bi )| =|Bi|}|G|. Therefore if wi is the number of nonprincipal characters of G giving a nonzero character sum over Bi then 2 2 2 2 a +m wi=a |G| for all i>1 and (a\m) +m w1=(a\m)|G|, so that 2 2 wi=a(|G|&a)Âm for all i>1 and w1=a(|G|&a)Âm &1\(|G|&2a)Âm. It follows that, for h>1, m divides |G| and m | a. Notice that we can construct a difference set or RDS by fixing one building block at a time. The character properties of the building blocks already fixed do not change as further building blocks are determined. This is an important advantage over the binary array viewpoint, in which we must consider the cross-correlation of a new array with each of those previously fixed. The remainder of the paper is concerned with describing and applying constructions for covering EBSs and BSs. In Section 3 we give a recursive

File: 582A 279612 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3283 Signs: 2706 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 25 construction for covering EBSs which assumes the existence of certain families of BSs. These families are provided by a recursive construction for BSs presented in Section 4. We shall illustrate the constructions of Sections 3 and 4 by reference to (16, 8, 5, +) covering EBSs on groups of order 64 and (32, 16, 11, &) covering EBSs on groups of order 128, which by Theorem 2.4 produce difference sets with parameters (320, 88, 24, 64) and (1408, 336, 80, 256) respectively. In Sections 5 and 6 we apply the recursive constructions and use Theorem 2.4 to provide a unifying framework for difference sets in the McFarland, Spence and Hadamard parameter families as well as the new parameter family (4). Section 5 deals with BSs on p-groups only, whereas Section 6 uses BSs on groups whose order need not be a prime power. In Section 7 we show that the construction of Section 4 is suf- ficiently strong to provide many further existence results for families of BSs. From Theorem 2.2 we deduce the existence of several families of semi- regular RDSs in Section 8 and provide a unifying framework for many previously known results on RDSs. We conclude in Section 9 with a selection of open problems and a discussion of how the definitions and constructions of this paper might be generalised to deal with nonabelian groups.

3. RECURSIVE CONSTRUCTION OF EXTENDED BUILDING SETS

In this section we give a recursive construction for covering EBSs. By Theorem 2.4, this central result implies a recursive construction for difference sets, which is the unifying construction of the title of the paper. We shall construct a covering EBS on a group G as the multiset union of two collections of building blocks. The first collection will be a (uam, um, h, \) EBS on G with respect to a subgroup U of order u.By the definition of EBS, the nonprincipal characters of G giving a nonzero character sum on these building blocks are precisely those which are principal on U. The second collection will be a (uam, um, t)BSonGrelative to U, where um=at (so the BS parameters can equivalently be written as (a2t, at, t).) By the definition of BS, the nonprincipal characters of G giving a nonzero character sum on these building blocks are precisely those which are nonprincipal on U. Moreover, since each building block of a BS or EBS has nonzero character sum for at most one nonprincipal character, the multiset union of these two collections will be a (uam, um, h+t,\) covering EBS on G. In this way we shall combine the favourable properties of the two collections of blocks without introducing unwanted interactions between them. We can fix |G|=u2am from the relationship between BS parameters given after Theorem 2.2. By Theorem 2.2, the second collection of building blocks can be viewed as a special form of (u2m2, u, u2m2, u2m) semi-regular

File: 582A 279613 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3267 Signs: 2869 . Length: 45 pic 0 pts, 190 mm 26 DAVIS AND JEDWAB

RDS relative to U in a group having G as a subgroup of index t. To form the first collection of building blocks we begin with a covering (am, m, h,\) EBS on the quotient group GÂU, which by Theorem 2.4 can be viewed as a special form of (uamh, m(ah\1), m(ah\1&m), m2)-difference set in a group having GÂU as a subgroup of index h. We now show how to take u copies of this covering EBS on GÂU to produce a EBS on G with respect to U, as required for the first collection of building blocks.

Lemma 3.1. Suppose there exists a (am, m, h,\) covering EBS on a group GÂU, where U is a subgroup of G of order u. Then there exists a (uam, um, h,\)EBS on G with respect to U.

Proof. Let [B$1, B$2, ..., B$h] be a (am, m, h, \) covering EBS on GÂU.

For each j let Bj=[g#G|gU # B$j] be the pre-image of B$j under the quotient mapping from G to GÂU. Since Bj is the union of |B$j | distinct cosets of U, it follows both that |Bj |=u |B$j| and that for every nonprincipal character / of G

0 if / nonprincipal on U /(Bj)= {u(B$j ) if / principal on U, where is the nonprincipal character induced by / on GÂU. By the definition of covering EBS, (B$j ) is nonzero (having modulus m) for exactly one value of j. Therefore [B1, B2, ..., Bh] is a (uam, um, h, \) EBS on G with respect to U. K

We can now construct a covering EBS on G from two ingredients: a covering EBS on GÂU (a special form of difference set) and a BS on G relative to U (a special form of semi-regular RDS). The following theorem is the key construction of the paper.

Theorem 3.2. Let G be a group of order u2am containing a subgroup U of order u. Suppose there exists a (am, m, h,\)covering EBS on GÂU and there exists a (a2 t, at, t) BS on G relative to U, where um=at. Then there exists a (uam, um, h+t,\)covering EBS on G.

Proof. By Lemma 3.1 the existence of a (am, m, h, \) covering EBS on

GÂU implies the existence of a (uam, um, h, \) EBS, say [B1, B2, ..., Bh], on G with respect to U. By assumption there exists a (a2t, at, t) BS, say

[Bh+1, Bh+2, ..., Bh+t],onGrelative to U. Since um=at the parameters of the BS can be written as (uam, um, t). By the definitions of EBS and BS, this implies that [B1, B2 , ..., Bh+t] is a (uam, um, h+t, \) covering EBS on G. K

File: 582A 279614 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 2920 Signs: 2207 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 27

By applying the relationship between covering EBS parameters given after Theorem 2.4 to GÂU, we find the parameters of Theorem 3.2 are constrained by am(amh\2m)=(amh\m&m2)|GÂU|, which reduces to ah(u&1)=muÃ(u&2). 2 3 5 For example, let G be the group Z4_Z2 , Z4_Z2 ,orZ2. In each case 2 3 G contains a subgroup U$Z2 such that GÂU$Z2 . Now there exists a tri- 3 vial (2, 1, 1, &) covering EBS [B$1] on Z2 , comprising just the identity element. Following the proof of Lemma 3.1, set B1=[g#G | gU # B$1]=U. Assume we can find a (8, 4, 2) BS [B2, B3] on G relative to U. (Section 2 2 contains an example of such a BS for the case G=Z4_Z2 .) Then by Theorem 3.2, [B1, B2, B3] is a (8, 4, 3, &) covering EBS on G and by

Theorem 2.4 there exists a (96, 20, 4, 16)-difference set in G_Z3 . In the above example B1 is a subgroup of G, but this need not be the 2 2 4 6 2 case. Take G to be the group Z4_Z2 , Z4_Z2 ,orZ2and let U$Z2 be a 4 4 subgroup of G such that GÂU$Z2 . Now we have seen in Section 1 that Z2 contains a (16, 6, 2, 4)-difference set, which can be viewed as a (4, 2, 1, +) 4 covering EBS [B$1] on Z2 . Again set B1=[g#G | gU # B$1], which is not a subgroup of G. Assume we can find a (16, 8, 4) BS [B2, B3, B4, B5] on

G relative to U. Then by Theorem 3.2, [B1, B2, B3, B4, B5] is a (16, 8, 5, +) covering EBS on G and by Theorem 2.4 there exists a (320, 88, 24,

64)-difference set in G_Z5 . These difference set parameters belong to the new family (4) with d=1, for which no examples of difference sets were previously known. As a further example, we show that the covering EBS on GÂU can com- prise more than one building block. Take G to be any group of order 128 2 and exponent at most 4, and let U$Z2 be a subgroup of G such that GÂU 3 5 is isomorphic to Z4_Z2 or Z2 . From the first example above, there is a (8, 4, 3, &) covering EBS on GÂU. Assume we can find a (32, 16, 8) BS on G relative to U. Then there exists a (32, 16, 11, &) covering EBS on G and therefore a (1408, 336, 80, 256)-difference set in G_Z11 . These difference set parameters belong to the McFarland family with q=4 and d=2, for which the only abelian groups previously known to contain dif- 7 5 ference sets were Z2_Z11 [42] and Z4_Z2_Z11 [20]. We shall show in Section 4 how to construct the BSs whose existence was assumed in the above examples. Following the pattern indicated by the examples, we now apply Theorem 3.2 recursively to large classes of groups, assuming for now that the required BSs are available.

Theorem 3.3. Let p be a prime, let r1, and for each d0 let Gd be a set of groups of order p(d+1)ram. Suppose there exists a (am, m, h,\)covering EBS on each G0 # G0 . Suppose also that, for each d1, there exists a

File: 582A 279615 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3386 Signs: 2701 . Length: 45 pic 0 pts, 190 mm 28 DAVIS AND JEDWAB

(d&1)r 2 (d&1)r (d&1)r ( p a t, p at, p t) BS on each Gd # Gd relative to a subgroup Ud r r (depending on Gd ) of order p , where p m=at and where Gd&1 contains a dr dr group isomorphic to Gd ÂUd . Then for each d0 there exists a ( p am, p m, dr r h+((p &1)Â(p &1))t,\)coverings EBS on each Gd # Gd . Proof. The proof is by induction on d. The case d=0 is true by assumption. Assume the case d&1 to be true. For each Gd # Gd , by assump- (d&1)r 2 (d&1)r (d&1)r tion there exists a ( p a t, p at, p t)BSonGd relative to a subgroup Ud . Since Gd&1 contains a group isomorphic to Gd ÂUd (of order pdram), by the inductive hypothesis there exists a ( p(d&1)ram, p(d&1)rm, (d&1)r r h+((p &1)Â(p &1))t, \) covering EBS on Gd ÂUd . It follows from Theorem 3.2 that the case d is true, completing the induction. K When applying the recursive construction for covering EBSs of Theorem (d+1)r 3.3 we shall usually take Gd to be the set of all p-groups of order p am with bounded exponent (independent of d). The condition that Gd&1 contains a group isomorphic to Gd ÂUd will then automatically be satisfied. In order to apply this theorem we require suitable families of BSs. We shall show how to obtain these in Section 4 by means of a second recursive construction. In Section 5 we shall deduce the existence of families of covering EBSs which, by Theorem 2.4, implies the existence of families of difference sets. In Section 6 we shall use a similar procedure to construct difference sets with parameters from the Hadamard family (1), applying Theorem 3.2 directly instead of Theorem 3.3.

4. RECURSIVE CONSTRUCTION OF BUILDING SETS

In this section we give a recursive construction for BSs relative to an elementary abelian subgroup. This will be used to provide the families of BSs needed for both the construction of difference sets in Sections 5 and 6 and for the construction of semi-regular RDSs in Section 8. The construction depends on a vector space P of dimension 2 over GF( pr). Let $ be a multiplicative generator of GF( pr). We can write the additive group of GF( pr)as($i |0ir&1) and so P is an additive group which can be written as (($i,0),(0,$j)|0i,jr&1). We con- 2r struct an isomorphism from P to Zp =(x1, x2, ..., xr , y1 , y2 , ..., yr) by i j ($ ,0)[xi+1 and (0, $ ) [ yj+1. The subspaces of P of dimension 1 (the r hyperplanes) H0 , H1 , ..., Hp r each contain p elements and have as their bases [(1, 0)], [(0, 1)], [(1, 1)], [($,1)],[($2,1)],[($3,1)], ..., [($ pr&2,1)].

Lemma 4.1. Let P be a vector space of dimension 2 over GF( pr), where p is prime. Any nonprincipal character of P is principal on exactly one of the hyperplanes of P.

File: 582A 279616 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3352 Signs: 2599 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 29

Proof. We show firstly that the kernels of the nonprincipal characters / of P are precisely the subgroups of P of order p2r&1. Since / is a homomorphism from P onto the pth roots of unity, |Ker(/)|=|P|Âp=p2r&1 and so Ker(/) is a subgroup of P of order p2r&1. Furthermore any subgroup of P of order p2r&1 is the kernel of some nonprincipal character of P. We next show that a subgroup of P of order p2r&1 contains at most one hyperplane. P is a vector space of dimension 2 over GF( pr) and each hyperplane is a subspace of dimension 1. Hence two distinct hyperplanes intersect in a subspace of dimension 0: the identity element. Therefore the product of two distinct hyperplanes is the whole of P, so a subgroup of order p2r&1 cannot contain two distinct hyperplanes. Finally we use a counting argument to show that a subgroup of P of 2r&1 order p contains exactly one hyperplane. Let Hi be a hyperplane for r 2r r some i=0, 1, ..., p . Since P$Zp we have PÂHi$Zp. Therefore PÂHi contains ( pr&1)Â(p&1) subgroups of order pr&1. Each such subgroup of 2r&1 PÂHi is associated with a subgroup of P of order p containing Hi , using the quotient mapping from P to PÂHi . Therefore there are at least r 2r&1 ( p &1)Â(p&1) distinct subgroups of P of order p containing Hi . Since i can take pr+1 values, there are at least ( pr+1)(pr&1)Â(p&1)= (p2r&1)Â(p&1) distinct subgroups of P of order p2r&1 containing some hyperplane (since we have already shown that the subgroups of P arising from different values of i must be distinct). But the total number of subgroups of P of order p2r&1 is ( p2r&1)Â(p&1) and so every subgroup of P of order p2r&1 contains exactly one hyperplane. We have now shown that for any nonprincipal character / of P, Ker(/) contains exactly one hyperplane of P. This completes the proof. K Lemma 4.1 implies the following result, due to Davis [10], which was discussed when introducing building sets in Section 2.

r r r 2r r Corollary 4.2. There exists a ( p , p , p ) BS on Zp relative to Z p , where p is prime and r1.

2r r Proof. Let H0 , H1 , ..., Hp r be the subgroups of Zp of order p corre- 2r sponding to hyperplanes of P under the isomorphism from Zp to P. Label the r subgroups so that H0=Z p. Then Lemma 4.1 implies that [H1 , H2, ..., Hp r] r r r 2r r is a ( p , p , p )BSonZp relative to Z p . K We now show how to exploit the hyperplane structure of Lemma 4.1 to obtain a more general result than Corollary 4.2. Take a group G containing 2r a subgroup Q isomorphic to Zp and consider those subgroups Hi of G which correspond to hyperplanes when viewed as subgroups of Q.We r show that if there exists a BS on GÂHi relative to QÂHi for i=1, 2, ..., p

File: 582A 279617 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3297 Signs: 2658 . Length: 45 pic 0 pts, 190 mm 30 DAVIS AND JEDWAB

then each BS can be ``lifted'' from the quotient group GÂHi to G (in a similar manner to the lifting of a covering EBS in Lemma 3.1) to collec- tively form a BS on G relative to H0 .

Theorem 4.3. Let G be a group of order p2ra containing a subgroup 2r Q$Zp , where p is prime. Let H0 , H1 , ..., Hp r be the subgroups of G of order pr corresponding to hyperplanes when viewed as subgroups of Q.

Suppose there exists a (a, - at, t) BS on GÂHi relative to QÂHi for each i= r r r r 1, 2, ..., p . Then there exists a ( p a, p - at, p t) BS on G relative to H0 .

Proof. For each i1, let [B$i1, B$i2, ..., B$it] be a (a, - at, t)BSonGÂHi relative to QÂHi . Following the proof of Lemma 3.1, for each i1 and for each j let Bij=[g#G | gHi # B$ij]. Since Bij is the union of |B$ij |=a distinct r cosets of Hi ,|Bij |=p a and for every nonprincipal character / of G and for each i1 and for each j

0 if / nonprincipal on Hi /(Bij)= r (5) {p (B$ij ) if / principal on Hi , where (B$ij ) is the nonprincipal character induced by / on GÂHi . By the definition of BS, for each i1, (B$ij ) is nonzero (having modulus - at ) for exactly one value of j if is nonprincipal on QÂHi , and is nonzero for no value of j if is principal on QÂHi . r r We claim that [Bij |1ip,1jt], comprising p t subsets Bij of G, r r r is a ( p a, p - at, p t)BSonGrelative to H0 . To establish this, let / be a nonprincipal character of G. Lemma 4.1 implies that if / is nonprincipal on Q then it is principal on one of the subgroups Hi and nonprincipal on all the others. We therefore distinguish three cases: / is principal on HI for some I{0 and nonprincipal on Hi for each i{I; / is principal on H0 and nonprincipal on Hi for each i{0; and / is principal on Q and nonprincipal on G.

In the first case, where / is principal on HI for some I{0 and nonprin- r cipal on Hi for each i{I, /(Bij )=0 for each i{I and /(BIj )=p (B$Ij ), from (5). Since / is nonprincipal on Q, is nonprincipal on QÂHI and so

(B$Ij ) is nonzero (having modulus - at ) for exactly one value of j. There- r fore /(Bij ) is nonzero (having modulus p - at ) for exactly one ordered pair (i, j ). In the second case, where / is principal on H0 and nonprincipal on Hi for each i{0, /(Bij )=0 for each ordered pair (i, j ), from (5). In the third case, where / is principal on Q and nonprincipal on G, / is principal r on Hi for each i0. Therefore /(Bij)=p (B$ij ) for each i1, from (5). Since is principal on QÂHi , (B$ij )=0 for each ordered pair (i, j ). The results for the three cases establish the claim. K

File: 582A 279618 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3325 Signs: 2493 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 31

r Given a group G and a subgroup H0$Z p on which we wish to construct a BS using Theorem 4.3, we are free to choose Q to be any subgroup of 2r G isomorphic to Zp containing H0 . This choice will determine the subgroups Hi{H0 of G corresponding to hyperplanes. By suitable choice of generators of G we can assume that Q is contained in 2r direct factors of

G and that any one particular hyperplane Hi is contained in r of these direct factors. Then the proof of Theorem 4.3 describes a procedure for constructing the BS explicitly. Given a BS on each of the pr quotient groups GÂHi relative to QÂHi , we lift each BS from GÂHi to G by taking r Bij=[g#G | gHi # B$ij]. This produces the p t building blocks of a r r r ( p a, p - at, p t)BSonGrelative to H0 . To illustrate this procedure in detail, suppose we wish to construct a 3 4 4 4 2 (32, 16, 8) BS on G=Z4_Z2=(x, y, z, w | x =y =z =w =1) relative 2 2 2 4 to H0=(x , y )$Z2. We firstly choose the subgroup Q$Z2 of G to be 2 2 2 (x , y , z , w), which contains H0 . We next determine the subgroups of G corresponding to hyperplanes, by reference to the multiplicative structure of GF(4). Since x2+x+1 is an irreducible polynomial of degree 2 over GF(2) we can regard GF(4) as having multiplicative generator $, where $2=$+1. Then the hyperplanes of GF(4)2 are ((1, 0)), ((0, 1)), ((1, 1)), (($,1)) and (($+1, 1)). Define the isomorphism from GF(4)2 to Q by (1, 0) [ x2,($,0)[y2,(0,1)[z2and (0, $) [ w. The subgroups 2 2 of G corresponding to the hyperplanes are then respectively H0=(x , y ), 2 2 2 2 2 2 2 2 2 2 2 2 H1=(z ,w), H2=(x z ,y w), H3=(y z ,x y w) and H4=(x y z ,x w). For each i{0 we now form the quotient group GÂHi and its associated 2 2 subgroup QÂHi . In this case we find that GÂHi$Z4_Z2 , and QÂHi$Z2 is 2 contained within Z4 , for each i{0. We therefore require a (8, 4, 2) BS on (a, b, c | a4=b4=c2) relative to (a2, b2). An example of such a BS was given in Section 2, comprising the group ring elements

2 2 3 3 B$1(a, b, c)=1+a+ac+a c+a bc+ab+ab c+b ,

3 3 2 2 2 3 3 2 B$2(a, b, c)=1+a +a b c+a b c+bc+ab c+ab +a b.

In order to construct the BS on G we write each quotient group GÂHi explicitly in terms of its generators. We find GÂH1=(xH1, yH1 , zH1), GÂH2=(xH2, yH2 , xzH2), GÂH3=(xH3, yH3 , yzH3) and GÂH4=

(xH4, yH4 , xyzH4), the first two generators having order 4 and the third 2 2 generator having order 2 in each case. We also find QÂHi$(x Hi, y Hi) for each i{0. Therefore a (8, 4, 2) BS in GÂHi relative to QÂHi is given by the building blocks B$i1 and B$i2 where for j=1, 2 we have

B$1j=B$j (x, y, z) H1 , B$2j=B$j (x, y, xz) H2 , B$3j=B$j (x, y, yz) H3 and B$4j= 2 3 3 B$j (x, y, xyz) H4 . For example, B$21=H2+xH2+x zH2+x zH2+x yzH2 2 3 3 +xyH2+x y zH2+y H2 . Each of the expressions B$ij is a group ring

File: 582A 279619 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3659 Signs: 2689 . Length: 45 pic 0 pts, 190 mm 32 DAVIS AND JEDWAB

element in Z[GÂHi ] comprising 8 elements of the quotient group GÂHi .We finally obtain Bij=[g#G | gHi # B$ij] by regarding the formal expression for B$ij as a group ring element in Z[G] comprising 32 elements of G. The

8 building blocks [Bij |1i4, 1j2] then form a (32, 16, 8) BS on G relative to H0 .

In general the quotient groups GÂHi for i{0 need not be isomorphic. 2 4 For example, let G=Z2_Z4=(x, y | x =y =1). The subgroups of G 2 2 corresponding to hyperplanes are H0=(x), H1=(y ) and H2=(xy ), 2 so the quotient groups GÂH1$Z2 and GÂH2$Z4 are not isomorphic. 2 (Since there exists a (2, 2, 2) BS on Z2 relative to Z2 but not on Z4 relative to Z2 , we cannot use Theorem 4.3 to construct a (4, 4, 4) BS on G relative to H0 .) This example also demonstrates (for i=2) that the direct factors of G containing H0 do not necessarily correspond to the direct factors of GÂHi containing QÂHi . Clearly we require some information about the form of GÂHi and QÂHi in order to apply Theorem 4.3 effectively. We now show that by appropriate choice of generators, exactly r direct factors of G retain the same exponent in GÂHi (these are the direct factors which contain QÂHi) and r are reduced by a factor of p.

2r Lemma 4.4. Let G be the group >u=1 Zp1+:u containing a subgroup 2r Q$Zp , where p is prime and :u0. Let H0 , H1 , ..., Hp r be the subgroups of G of order pr corresponding to hyperplanes when viewed as subgroups of

Q. Then for each Hi there exists a r-element subset S of [1, 2, ..., 2r] such that GÂHi$>u Â S Zp 1+:u_>u # S Zp:u . Moreover, for each Hi a suitable r choice of generators of G ensures that QÂHi$Zp is contained in the first r direct factors of GÂHi as specified. Furthermore if H0 is contained in a sub- r group of G isomorphic to Zp 2 then, for each Hi{H0 , QÂHi is contained in r a subgroup of GÂHi isomorphic to Zp2 .

r r Proof. Each Hi is a subgroup of Q of order p and so Hi$Zp . There- fore we can choose generators of G such that Hi is contained in r direct p 1+:u factors of G. Let [xu |1u2r] be the generators of G, where xu =1 for all u, and let S be the r-element subset of [1, 2, ..., 2r] which indexes p:u the r direct factors containing Hi . Then Hi=(xu | u#S), and the 1+:u :u order of xuHi in GÂHi is p for u Â S and p for u # S. Therefore

GÂHi=(xuHi |1u2r)$>u Â S Zp1+:u_>u # S Zp:u as required. With p:u p:u this choice of generators, QÂHi=(xu Hi |1u2r)=(xu Hi | uÂS), r and so QÂHi$Z p and QÂHi is contained in the first r direct factors of GÂHi as specified. r Suppose now that H0 is contained in a subgroup of G isomorphic to Zp 2 . r Since H0 is isomorphic to Z p this is equivalent to the statement that each p h0 # H0 can be written as h0=g for some g # G. For any Hi{H0 let qHi

File: 582A 279620 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3567 Signs: 2671 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 33

be an element of QÂHi . The proof of Lemma 4.1 shows that Q=H0 Hi and p so q=h0 hi for some h0 # H0 and some hi # Hi . Since h0=g for some g # G p p p r we obtain qHi=g hi Hi=g Hi=(gHi ) , and so QÂHi$Zp is contained in r a subgroup of GÂHi isomorphic to Zp 2 . K

For example, we can now construct the BSs whose existence we assumed in Section 3. We firstly show there exists a (8, 4, 2) BS on each of the 2 3 5 2 groups Z4_Z2 , Z4_Z2 and Z2 relative to a subgroup U$Z2 contained 2 within two of the largest direct factors of the group. The group Z4_Z2 is dealt with by the example in Section 2. For the other two groups, by 4 2 Corollary 4.2 there is a (4, 4, 4) BS on Z2 relative to Z2 , from which the desired BS can be obtained using Lemma 2.1 with s=2. An example in Section 3 then shows that there is a (96, 20, 4, 16) McFarland difference set in any group of order 96 whose Sylow 2-subgroup has exponent at most 4. Next we show there exists a (32, 16, 8) BS on any group G of order 128 2 and exponent at most 4 relative to a subgroup U$Z2 contained within 4 two of the largest direct factors of G. Let Q$Z2 be a subgroup of G containing H0=U. By Lemma 4.4, for each Hi{H0 , we find GÂHi has order 2 32 and exponent at most 4 and QÂHi$Z2 is contained in two of the largest direct factors of GÂHi . By the preceding example there is a (8, 4, 2) BS on GÂHi relative to QÂHi and so by Theorem 4.3 we obtain the desired BS on G. An example in Section 3 then shows that there is a (1408, 336, 80, 256) McFarland difference set in any group of order 1408 whose Sylow 2-subgroup has exponent at most 4. The above procedure indicates the recursive construction for McFarland difference sets which we shall present in Section 5. As a further example, we show there exists a (16, 8, 4) BS on each of the 2 2 4 6 2 groups Z4_Z2 , Z4_Z2 and Z2 relative to a subgroup U$Z2 contained within two of the largest direct factors of the group. We firstly consider the 2 2 3 3 group Z4_Z2. Now Jungnickel [28] has shown that [1, x, y, x y ] is a 2 4 4 (4, 4, 4, 1) semi-regular RDS in Z4=(x, y | x =y =1) relative to 2 2 2 2 (x , y )$Z2. Since this is equivalent to a (4, 2, 1) BS on Z4 relative to 2 Z2 , we obtain the required BS by Theorem 4.3 and Lemma 4.4. We cannot 3 5 deal with the groups Z4_Z2 and Z2 in the same way because this would 2 4 2 require a (4, 4, 4, 1) semi-regular RDS on Z4_Z2 or Z2 relative to U$Z2 , which does not exist [24]. But from Corollary 4.2 there is a (8, 8, 8) BS 6 3 on Z2 relative to Z2 . We show in the following lemma how this can be used 5 2 to provide a (8, 8, 8) BS on Z2 relative to Z2 , which allows the required BS to be constructed using Lemma 2.1 with s=2. An example in Section 2 2 3 then shows that there is a (320, 88, 24, 64)-difference set in Z4_Z2_Z5 , 4 6 3 Z4_Z2_Z5 and Z2_Z5 . Although the group Z4_Z5 has order 320 and 3 exponent 4 it is excluded from this result because Z4 does not contain a

File: 582A 279621 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3551 Signs: 2874 . Length: 45 pic 0 pts, 190 mm 34 DAVIS AND JEDWAB

4 subgroup Q$Z2 and so Theorem 4.3 cannot be applied. In Section 5 we present a recursive construction for difference sets with parameters (4) 3 based on these initial examples, and show that the exceptional case Z4_Z5 does not propagate to larger groups under this construction. We now describe the method of ``contraction'' of BSs required in the preceding example, modelled on that given by Elliott and Butson [21] for RDSs.

Lemma 4.5. Suppose there exists a (a, - at, t) BS on a group G relative to a subgroup U. Let W be a subgroup of U. Then there exists a (a, - at, t) BS on GÂW relative to UÂW.

Proof. Let [B1, B2..., Bt] be a (a, - at, t)BSonGrelative to U. Let

B$j=[gU#GÂU | g # Bj] be the image of Bj under the quotient mapping from G to GÂU and let B"=j [gW#GÂW | g # Bj] be the image of Bj under the quotient mapping from G to GÂW.

We show firstly that for each j, B$j=GÂU in the group ring Z[GÂU]. Every nonprincipal character of GÂU can be regarded as being induced by a character / that is nonprincipal on G and principal on U, so that

(B$j )=/(Bj )=0 by the definition of BS. Therefore B$j=c GÂU in Z[GÂU] for some integer c. Now |Bj |=a by the definition of BS, and |G|Â|U|=a from the relationship between BS parameters given after Theorem 2.2, so that c=1.

We next show that for each j, B"j contains no repeated elements. Since

B$j=GÂU in Z[GÂU], each coset of U in G contains exactly one element of Bj . It follows that each coset of W in G contains at most one element of Bj . Finally we show that [B"1, B"2..., Bt"] is a (a, - at, t)BSonGÂW relative to UÂW. For each j, we have shown that B"j is a subset of GÂW comprising a elements. Every nonprincipal character , of GÂW can be regarded as being induced by a character / that is nonprincipal on G and principal on

W, so that ,(Bj")=/(Bj ). If / is nonprincipal on U, so that , is nonprincipal on UÂW, then /(Bj) is nonzero (having modulus - at ) for exactly one j, by the definition of BS. If / is principal on U, so that , is principal on UÂW, then /(Bj )=0 for each j, by the definition of BS. This completes the proof. K

We shall defer until Section 7 a full examination of the consequences of Theorem 4.3 for constructing families of BSs. For now our goal is to obtain quickly and easily just the BSs required for the construction of difference sets using Theorems 2.4 and 3.3. In this spirit we now give a recursive application of Theorem 4.3 to large classes of groups, which will be generalised in Section 7. While it was not important in Theorem 3.3 to

File: 582A 279622 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3097 Signs: 2529 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 35 keep track of the subgroup U associated with the BS on the group G, here we specify ordered pairs (G, U) to assist in applying Theorem 4.3 recursively.

Theorem 4.6. Let p be prime, let r1, and for each d1 let Kd be a dr set of ordered pairs (Gd , Ud ), where Gd is a group of order p a containing r a subgroup Ud$Z p . Suppose that for each (G1 , U1 )#K1 there exists a (a, - at, t) BS on G1 relative to U1 . Suppose also that, for each d>1 and 2r for each (Gd , Ud )#Kd, Gd contains a subgroup Q$Zp and subgroups r H0 , H1 , ..., Hp r of order p (corresponding to hyperplanes when viewed as subgroups of Q), where H0=Ud and where Kd&1 contains an ordered pair isomorphic to (Gd ÂHi , QÂHi) for each Hi{H0 . Then for each d1 and for (d&1)r (d&1)r (d&1)r each (Gd , Ud )#Kd there exists a ( p a, p -at, p t) BS on Gd relative to Ud . Proof. The proof is by induction on d. The case d=1 is true by assumption. Assume the case d&1 to be true and consider Hi{H0 . Since

Kd&1 contains an ordered pair isomorphic to (Gd ÂHi , QÂHi ), by the induc- (d&2)r (d&2)r (d&2)r tive hypothesis there exists a ( p a, p -at, p t)BSonGdÂHi relative to QÂHi . It follows from Theorem 4.3 that the case d is true, completing the induction. K

In Section 5 we shall apply Theorem 4.6, usually taking Kd to be the set dr of all ordered pairs (Gd , Ud ) for which Gd is a p-group of order p a with r bounded exponent (independent of d), and for which Ud$Zp is contained in r of the largest direct factors of Gd . This will allow the construction of difference sets with parameters from the families (2), (3) and (4). In Sec- tion 6 we shall present a different recursive application of Theorem 4.3 for the construction of difference sets with parameters from the Hadamard family (1).

5. APPLICATION TO DIFFERENCE SETS

In this section we use the recursive construction of Theorem 4.6 to obtain families of BSs on p-groups, from which the recursive construction of Theorem 3.3 produces families of covering EBSs on p-groups. Using Theorem 2.4 we deduce the existence of difference sets with parameters from the McFarland family, the Spence family, and the new parameter family (4). We firstly show that by restricting the BSs to be on p-groups, the resulting difference set parameters must belong to the Hadamard family (1), the McFarland family (2), the Spence family (3), or the new family (4). By Theorem 2.4, the construction of Theorem 3.2 can be viewed as using an

File: 582A 279623 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3140 Signs: 2416 . Length: 45 pic 0 pts, 190 mm 36 DAVIS AND JEDWAB initial difference set (based on the covering EBS on GÂU) to produce a final difference set (based on the covering EBS on G). We now examine the parameters u, a, m, t and h of Theorem 3.2 in more detail. By assumption G is a p-group of order u2am,souand m are powers of the prime p. Let u=pr, and write m=p(d&1)r+: for some d1 and 0:

File: 582A 279624 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3413 Signs: 2920 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 37 we require a (1, 1, 2, &) covering EBS on a group of order pr and a (2, 1, 1, &) covering EBS on a group of order 2r+1 respectively. Similarly, in the second case we require a (3d,3d,3d) BS on a group of order 3d+1 relative to a subgroup of order 3 and, to begin the recursion with d=1, a (1, 1, 1, +) covering EBS on a group of order 3. Likewise in the third case we have a=1 or a=2. When a=1 we require a(22d+1,22d+1,22d+1) BS on a group of order 22d+3 relative to a subgroup of order 4 and, for d=1, a (2, 2, 2, +) covering EBS on a group of order 8. When a=2 we require a (22d+2,22d+1,22d) BS on a group of order 22d+4 relative to a subgroup of order 4 and, for d=1,a(4,2,1,+) covering EBS on a group of order 16. In each case it is straightforward to find an appropriate covering EBS, so the application of the recursive construction depends on finding families of BSs as described above. We now show how to construct some of the identified families using the recursive construction for BSs of Theorem 4.6. The initial BSs can be traced to three sources: the ( pr, pr, pr)BSof Corollary 4.2, due to Davis [10]; the (8, 4, 2) BS described in Section 2, 2 2 due to Arasu and Sehgal [3]; and the (4, 2, 1) BS on Z4 relative to Z2 described in Section 4, due to Jungnickel [28].

Theorem 5.1. For each d1, the following exist:

dr dr dr (d+1)r r (i) A (p , p , p ) BS on Zp relative to Z p , where p is prime and r1. 2d+1 2d 2d&1 2d+3 (ii) A (2 ,2 ,2 ) BS on any group Gd of order 2 and 2 exponent at most 4 relative to a subgroup Ud$Z2 contained within two of the largest direct factors of Gd . 2d+2 2d+1 2d 2d+4 (iii) A (2 ,2 ,2 )BS on any group Gd of order 2 and 2 exponent at most 4 relative to a subgroup Ud$Z2 contained within two of 3 the largest direct factors of Gd , except possibly G1=Z4 .

Proof. The proof is by application of Theorem 4.6, using initial BSs introduced in earlier sections. r (d+1)r r r For (i), put a=t=p and take Kd=[(Z p , Zp)], where H0=Zp is (d+1)r r r r contained within r direct factors of Zp . There exists a ( p , p , p )BS 2r r 2r on Zp relative to the subgroup Z p contained within r direct factors of Z p , 2r (d+1)r by Corollary 4.2. For d>1, let Q$Zp be a subgroup of Zp containing (d+1)r r H0 . For each subgroup Hi{H0 of Z p of order p , corresponding to a hyperplane when viewed as a subgroup of Q, Lemma 4.4 shows that (d+1)r dr r dr Zp ÂHi$Z p , and QÂHi$Zp is contained within r direct factors of Z p . (d+1)r Therefore Kd&1 contains an ordered pair isomorphic to (Zp ÂHi , QÂHi ) and the result follows from Theorem 4.6.

File: 582A 279625 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3201 Signs: 2545 . Length: 45 pic 0 pts, 190 mm 38 DAVIS AND JEDWAB

For (ii), put p=r=2, a=8 and t=2, and take Kd to be the set of all 2d+3 ordered pairs (Gd , Ud ) for which Gd is a group of order 2 and expo- 2 nent at most 4 and Ud$Z2 is a subgroup contained within two of the largest direct factors of Gd . An example in Section 4 shows that there exists 2 3 5 a (8, 4, 2) BS on each of the groups Z4_Z2 , Z4_Z2 and Z2 relative to a 2 subgroup Z2 contained within two of the largest direct factors of the group. 4 For d2, let Q$Z2 be a subgroup of Gd containing H0=Ud . For each subgroup Hi{H0 of Gd corresponding to a hyperplane, Lemma 4.4 shows 2d+1 that Gd ÂHi is a group of order 2 and exponent at most 4 and 2 QÂHi$Z2 is contained in two of the largest direct factors of Gd ÂHi . For (iii), put p=r=2, a=16 and t=4, and take Kd to be the set of all ordered pairs (Gd , Ud ) included in the statement of the theorem (so that 3 (Z4 , U1) ÂK1). An example in Section 4 shows that there exists a (16, 8, 4) 2 2 4 6 BS on each of the groups Z4_Z2 , Z4_Z2 and Z2 relative to a subgroup 2 Z2 contained within two of the largest direct factors of the group. The remainder of the proof is similar to that of (ii) except that we must ensure 3 4 G2 ÂHi$% Z4 for Hi{H0 . We achieve this by taking Q$Z2 to be a subgroup of Gd containing H0=Ud for d>1 as before, with the additional constraint that Q be contained within four of the largest direct factors of Gd . K 3 Note that although the group Gd=Z4 is not covered by the case d=1 of Theorem 5.1(iii), this exception does not propagate to higher values of d under the recursive construction of Theorem 4.6. We next combine the BSs of Theorem 5.1 with initial covering EBSs whose parameters were previously identified in order to produce families of covering EBSs.

Theorem 5.2. For each d0, the following exist: (i) A (pdr, pdr,(p(d+1)r&1)Â(pr&1)+1, &) covering EBS on (d+1)r Zp , where p is prime and r1. (ii) A (22d+1,22d,(22d+1+1)Â3, &) covering EBS on any group of order 22d+3 and exponent at most 4. d d d+1 d+1 (iii) A (3 ,3,(3 &1)Â2, +) covering EBS on Z3 . (iv) A (22d+2,22d+1,(22d+2&1)Â3, +) covering EBS on any group of 2d+4 3 order 2 and exponent at most 4, except possibly Z4 in the case d=1. Proof. The proof is by application of Theorem 0recurcebs to the BSs of Theorem 0diffsetbs, together with appropriate initial covering EBSs. r (d+1)r For (i), put a=m=1, h=2 and t=p , and take Gd=[Zp ] in r Theorem 3.3. There exists a trivial (1, 1, 2, &) covering EBS on Zp . The (d+1)r r required BSs are provided by Theorem 5.1(i), and Zp ÂZp is isomorphic dr to Zp , which is contained in Gd&1.

File: 582A 279626 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3329 Signs: 2540 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 39

For (ii), put p=r=2, a=t=2 and m=h=1, and take Gd to be the set of all groups of order 22d+3 and exponent at most 4. There exists a trivial 3 (2, 1, 1, &) covering EBS on Z4_Z2 and Z2 . The required BSs on Gd relative to Ud are provided by Theorem 5.1(ii), and clearly Gd ÂUd is a group of order 22d+1 and exponent at most 4. d+1 For (iii), put p=3, r=1, a=m=h=1 and t=3, and take Gd=[Z3 ]. There exists a (1, 1, 1, +) covering EBS on Z3 comprising two elements

(the complement of a trivial (2, 1, 1, &) covering EBS on Z3). The required d+1 BSs are provided by Theorem 5.1(i) with p=3 and r=1, and Z 3 ÂZ3 is d isomorphic to Z3 . For (iv), put p=r=2, a=m=2, h=1 and t=4, and take Gd to be the 2d+4 3 set of all groups of order 2 and exponent at most 4 but exclude Z4 from G1 . The examples of (16, 6, 2, 4)-difference sets in Section 1 are equiv- 2 2 4 alent to a (4, 2, 1, +) covering EBS on Z4 , Z4_Z2 and Z2 . The required 2d+2 BSs are provided by Theorem 5.1(iii), and Gd ÂUd is a group of order 2 and exponent at most 4. Furthermore the choice of Ud ensures that 3 G2 ÂU2$% Z4 . K We now list the families of difference sets arising from the covering EBSs of Theorem 5.2. The unifying corollary which follows is one of the central results of the paper. There are no abelian groups known to contain difference sets in the indicated families which are not covered by this result. (As throughout the paper, the groups involved are implicitly abelian.)

Corollary 5.3. For each d0, the following exist: (i) A McFarland difference set with q=pr in any group of order d+1 d+1 (d+1)r q ((q &1)Â(q&1)+1) containing a subgroup isomorphic to Zp , where p is prime and r1. (ii) A McFarland difference set with q=4 in any group of order 22d+3((22d+1+1)Â3) containing a subgroup of order 22d+3 and exponent at most 4. (iii) A Spence difference set in any group of order 3d+1((3d+1&1)Â2) d+1 containing a subgroup isomorphic to Z3 . (iv) A difference set with parameters (4) in any group of order 22d+4((22d+2&1)Â3) containing a subgroup of order 22d+4 and exponent at 3 most 4, except possibly when the subgroup is Z4 in the case d=1. Proof. Apply Theorem 2.4 to the covering EBSs of Theorem 5.2. K

In Corollary 5.3(i), the Sylow p-subgroup of the group containing the (d+1)r McFarland difference set is isomorphic to Zp when p is odd, and is (d+1)r+1 (d+1)r&1 isomorphic to Z2 or Z4_Z2 when p=2, because p divides

File: 582A 279627 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3084 Signs: 2373 . Length: 45 pic 0 pts, 190 mm 40 DAVIS AND JEDWAB the index (qd+1&1)Â(q&1)+1 if and only if p=2. In the remaining parts of Corollary 5.3, the Sylow p-subgroup of the group containing the difference set is isomorphic to the subgroup mentioned (where p=2 in parts (ii) and (iv) and p=3 in part (iii).) Corollary 5.3(i) is due to McFarland [42] for p odd and to Dillon [20] for p=2. Ma and Schmidt [39] showed that for p odd, the condition that (d+1)r the Sylow p-subgroup is isomorphic to Zp is necessary as well as sufficient, provided that p is self-conjugate modulo the group exponent. (The definition of self-conjugate is given before Lemma 1.1.) In a subsequent paper Ma and Schmidt [38] showed that for p=2 and r>1, the Sylow 2-subgroup must have exponent at most 4 provided that 2 is self-conjugate modulo the group exponent. (For p=2 and r=1 the McFarland parameters correspond to Hadamard parameters with N=2d, which are considered separately in Section 6.) Corollary 5.3(ii) extends the set of groups known to contain McFarland difference sets in the case q=4 beyond those identified in Corollary 5.3(i). None of these additional groups was previously known to contain difference sets with the single exception, due to Arasu and Sehgal [3], of 2 Z4_Z2_Z3 in the case d=1. Moreover the self-conjugacy condition from Ma and Schmidt's result [38] above is always satisfied when q=4, since the exponent of a group divides the order and 2 is self-conjugate modulo 22d+3((22d+1+1)Â3). We have therefore established that a McFarland difference set with q=4 exists in an abelian group if and only if the Sylow 2-subgroup has exponent at most 4. Unlike the above result for McFarland difference sets with q=pr odd, this result does not depend on a self- conjugacy condition. The only other comparable result for families of difference sets, relying on a single group exponent condition, is due to Kraemer [31] for Hadamard difference sets. We shall show in Section 6 that Kraemer's result can also be derived from the framework of this paper. Corollary 5.3(iii) is due to Spence [54]. Using Theorem 5.4 below it is easily shown that the condition that the Sylow 3-subgroup is isomorphic to d+1 Z3 is necessary as well as sufficient, provided that 3 is self-conjugate modulo the group exponent. Corollary 5.3(iv) describes the first new family of difference set parameters to be discovered since 1977 [54]. Apart from the case d=0, giving Hadamard parameters, all the examples were previously unknown. This also gives a new family of symmetric designs with the same parameters (4). For d=1, Ma and Schmidt [38] have shown that the Sylow 2-subgroup must have exponent at most 4. The only open case for 3 difference set parameters (320, 88, 24, 64) is therefore Z4_Z5 . For d>1, we can use standard techniques to bound the exponent of the Sylow 2-subgroup. We shall use the following special case of Theorem 4.33 of Lander [32], based on results of Turyn [55].

File: 582A 279628 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3306 Signs: 2909 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 41

Theorem 5.4. Suppose that there exists a (v, k, *, n)-difference set in an abelian group G containing a subgroup H of index w. Suppose also that p is a prime for which p | w, p2r | n for some r1, p is self-conjugate modulo exp(GÂH), and the Sylow p-subgroup of GÂH is cyclic. Then prwv.

Corollary 5.5. The Sylow 2-subgroup of a group containing a difference set with parameters (4) has exponent at most 8 provided that 2 is self-conjugate modulo the group exponent. Proof. The group G containing the difference set has order v= 22d+4((22d+2&1)Â3). Let the exponent of the Sylow 2-subgroup be 2:. Choose the subgroup H so that w=2:((22d+2&1)Â3) with the Sylow 2-subgroup of GÂH cyclic. Apply Theorem 5.4 with p=2 and r=2d+1 to obtain :3. K The discussion at the beginning of this section identifies the possible parameters for BSs on p-groups which could be used in Theorem 3.2 to produce difference sets. Not all of the identified parameter sets are included in Theorem 5.1. In particular, we have seen that a (2qd, qd, qdÂ2) BS on a group of order 2qd+1 relative to a subgroup of order q=2r would yield a McFarland difference set with parameters q and d (assuming the appropriate covering EBS existed). However the only groups on which we (d+1)r+1 (d+1) r&1 know that such BSs exist for q>4 are Z2 and Z4_Z 2 , using Lemma 2.1 on Theorem 5.1(i). Construction of such BSs on other groups would give new McFarland difference sets. For example, in the case q=8 and d=1, a (16, 8, 4) BS on a group of order 128 and exponent 4 5 (other than Z4_Z2) relative to a subgroup of order 8 would give a new (640, 72, 8, 64) McFarland difference set. The results of this paper grew out of an unsuccessful attempt to construct such a BS. In the case q=16 and d=1, a (32, 16, 8) BS on a group of order 512 and exponent 4 (other 7 than Z4_Z2) relative to a subgroup of order 16 would give a new (4608, 272, 16, 256) McFarland difference set. (We have imposed exponent 4 in both cases because the self-conjugacy condition from Ma and Schmidt's result [38] is always satisfied when d=1.)

6. APPLICATION TO HADAMARD DIFFERENCE SETS

In this section we use the key constructions, Theorem 3.2 for covering EBSs and Theorem 4.3 for BSs, to obtain difference sets with parameters from the Hadamard family (1). Although many of the results were previously known our intention is to show that the various construction methods in the literature can be concisely brought into the unifying

File: 582A 279629 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 2994 Signs: 2453 . Length: 45 pic 0 pts, 190 mm 42 DAVIS AND JEDWAB framework of this paper. Based on this formulation, we suggest a number of generalisations in Section 9. Since the Hadamard parameters with N=2d are equivalent to the McFarland parameters with q=2, we have already established in Corollary 5.3(i) that there exists a Hadamard difference set in any group of order 22d+2 and rank at least d+1. This construction depended on the number of building blocks h and t in Theorem 3.2 being large. However we now demonstrate by means of an example that for the Hadamard parameters it is sufficient to take h=t=2, which allows additional freedom in choosing the group G (which need not be a 2-group). 2 Assume we can find a (9, 3, 2, &) covering EBS on Z2_Z3 , so that by 2 2 Theorem 2.4 there exists a (36, 15, 6, 9) Hadamard difference set in Z2_Z3 2 2 2 and Z4_Z3. Assume also that we can find a (18, 6, 2) BS on Z2_Z3 relative to Z2 . By Theorem 3.2 with u=2, this BS and covering EBS 2 2 together give a (18, 6, 4, &) covering EBS on Z2_Z3 . Therefore by Theorem 2.4 there exists a (144, 66, 30, 36) Hadamard difference set in 2 G_Z3 for any group G of order 16 and exponent at most 8. Now apply Lemma 2.1 with s=2 to the (18, 6, 2) BS to obtain a 2 3 2 (36, 6, 1) BS on Z4_Z2_Z3 and Z2_Z3 relative to any subgroup of order 2. Then by Theorem 4.3 with pr=2 there exists a (72, 12, 2) BS on 2 G_Z3 relative to any subgroup of order 2, where G is any group of order 3 16 and exponent at most 4 (since GÂHi$Z4_Z2 or Z2). Furthermore we can apply Lemma 2.3 with s=2 to the (18, 6, 4, &) covering EBS to obtain 2 3 2 a (36, 6, 2, &) covering EBS on Z4_Z2_Z3 and Z2_Z3 . Then by Theorem 3.2 with u=2, these (36, 6, 2, &) covering EBSs together with the 2 (72, 12, 2) BSs give a (72, 12, 4, &) covering EBS on G_Z3 , where G is any group of order 16 and exponent at most 4. Therefore by Theorem 2.4 2 there exists a (576, 276, 132, 144) Hadamard difference set in G_Z3 for any group G of order 64 and exponent at most 16. The pattern indicated by this example forms the model for the constructions in this section, each step of the recursion using an initial Hadamard difference set with N=2d&1m to construct a final Hadamard difference set with N=2dm, where m is odd. Provided the initial BS and covering EBS 2 2 can be found, the group of order m (Z3 in the above example) plays no part in the recursion. For example, given a (1, 1, 2, &) covering EBS on 2 Z2 (which is trivial) and a (2, 2, 2) BS on Z2 relative to Z2 (which exists by Corollary 4.2), by the same argument as above there exists a (64, 28, 12, 16) Hadamard difference set in any group of order 64 and exponent at most 16. As in Theorem 3.3 we begin by assuming the existence of an initial covering EBS and a family of BSs.

Theorem 6.1. Let M be a group of odd order m2 and for each d1 let 2d d Gd be the set of all groups of order 2 and exponent at most 2 . Suppose that

File: 582A 279630 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3394 Signs: 2819 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 43

2 there exists a (m , m,2,&)covering EBS on Z2_M. Suppose also that for 2d&1 2 d each d1 and for each Gd # Gd there exists a (2 m ,2 m,2) BS on Gd_M relative to a subgroup Ud (depending on Gd ) of order 2. Then for 2d&1 2 d each d1 and for each Gd # Gd there exists a (2 m ,2m,4,&)covering EBS on Gd_M. Proof. The proof is by induction on d. We begin by establishing the case d=1. By assumption there exists a (m2, m, 2, &) covering EBS on 2 2 Z2_M and a (2m ,2m,2)BSon Z2_M relative to Z2 . Combine these 2 using Theorem 0master with G=Z2_M and U=Z2 to obtain the case d=1.

Assume the case d&1 to be true. For each Gd # Gd , by assumption there 2d&1 2 d exists a (2 m ,2 m,2)BSon Gd_M relative to a subgroup Ud of 2d&1 d order 2. Now Gd ÂUd has order 2 and exponent at most 2 and so d attains the exponent 2 in at most one direct factor. Therefore Gd ÂUd con- d&1 tains a subgroup SÂUd of index 2 and exponent at most 2 . The inductive hypothesis then implies that there exists a (22d&3m2,2d&1m, 4, &) covering 2d&2 2 EBS on (SÂUd)_M. Apply Lemma 2.3 with s=2 to obtain a (2 m , d&1 2 m, 2, &) covering EBS on (Gd ÂUd)_M. Combine this covering EBS with the BS on Gd_M relative to Ud , using Theorem 3.2 with G=Gd_M. This shows the case d is true and completes the induction. K We next show that the family of BSs required in Theorem 6.1 can be obtained recursively from a single BS using Theorem 4.3. Although

Theorem 6.1 requires a BS on Gd_M relative to only a single subgroup

Ud , the recursion produces a BS on Gd_M relative to any subgroup of order 2. By rewriting the generators of Gd we can assume that such a subgroup is contained within a single direct factor of Gd .

Theorem 6.2. Let M be a group of odd order m2 and for each d1 let 2d d Gd be the set of all groups of order 2 and exponent at most 2 . Suppose 2 2 there exists a (2m ,2m,2)BS on Z2_M relative to Z2 . Then for each d1 2d&1 2 d and for each Gd # Gd there exists a (2 m ,2m,2)BS on Gd_M relative to any subgroup of order 2. Proof. The proof is by induction on d. The case d=1 is true by assumption. Assume the case d&1 to be true. For each Gd # Gd , let H0 be 2 any subgroup of order 2. Choose Q$Z2 to be a subgroup of Gd containing H0 and let the subgroups of Gd of of order 2, corresponding to hyperplanes when viewed as subgroups of Q,beH0,H1 and H2 . For each i, Gd ÂHi has order 22d&1 and exponent at most 2d and so, as in the proof of

Theorem 6.1, Gd ÂHi contains a subgroup SÂHi of index 2 and exponent at most 2d&1. Then by the inductive hypothesis there exists a (22d&3m2, d&1 2 m,2) BSon(SÂHi)_M relative to QÂHi$Z2 . Apply Lemma 2.1

File: 582A 279631 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3259 Signs: 2585 . Length: 45 pic 0 pts, 190 mm 44 DAVIS AND JEDWAB

2d&2 2 d&1 with s=2 to obtain a (2 m ,2 m,1)BSon(GdÂHi)_M relative to 2d&1 2 d QÂHi . Therefore by Theorem 4.3 there exists a (2 m ,2m,2)BSon Gd_M relative to H0 . This shows the case d is true and completes the induction. K Previously, in Theorem 3.2, we gave a construction for a covering EBS based on the existence of a BS and another covering EBS. We now show how to construct a particular type of BS (required as the initial BS in Theorem 0hadbsfam) from a covering EBS.

Lemma 6.3. Let M be a group of odd order m2. Suppose there exists a 2 2 (m , m,2,&)covering EBS on Z2_M. Then there exists a (2m ,2m,2)BS 2 on Z2_M relative to Z2 .

2 2 2 2 Proof. Let Z2=(x, y | x =y =1) and let [A, B] be a (m , m,2,&) covering EBS on G=(y)_M, where the building block containing m2&m elements is A. Define the subsets C=A+x(G"A) and D=B+ 2 x(G"B)ofZ2_M. Let / be a nonprincipal character of (x)_G. Firstly consider the case when / is nonprincipal on G. By the definition of covering EBS, [|/(A)|, |/(B)|]=[0, m] so [|/(C)|, |/(D)|]=[0, m|1&/(x)|]. There- fore if / is also nonprincipal on (x) (so / maps x to &1) we have [|/(C)|, |/(D)|]=[0, 2m] whereas if / is principal on (x) then /(C)= /(D)=0. Next consider the case when / is principal on G (and so nonprincipal on (x)). This gives /(C)=|A|&(|G|&|A|)=&2m and /(D)= |B|&(|G|&|B|)=0. Combining the two cases, [C, D] is a (2m2,2m,2) BSon (x)_G relative to (x). K For example, at the beginning of this section we assumed the existence 2 2 2 of a (9, 3, 2, &) covering EBS on Z2_Z3 and a (18, 6, 2) BS on Z2_Z3 relative to Z2 . By Lemma 6.3 the existence of the second is implied by the existence of the first. We now combine Theorems 6.1 and 6.2 and Lemma 6.3 to show that only an initial covering EBS is required for the recursions.

Corollary 6.4. Suppose there exists a (m((m&1)Â2), m,4,+)covering EBS on a group M of odd order m2. Then the following exist:

2d&1 2 d (i) A (2 m ,2 m,2) BS on Gd_M relative to any subgroup 2d of order 2, where d1 and Gd is any group of order 2 and exponent at most 2d. 2d&1 2 d (ii) A (2 m ,2 m,4,&) covering EBS on Gd_M, where d1 2d d and Gd is any group of order 2 and exponent at most 2 .

File: 582A 279632 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 2807 Signs: 2124 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 45

d (iii) A Hadamard difference set with N=2 minGd_M,where d0 2d+2 d+2 and Gd is any group of order 2 and exponent at most 2 .

Proof. For (i), by assumption there exists a (m((m&1)Â2), m,4,+) covering EBS on M. Apply Lemma 2.3 with s=2 to obtain a (m(m&1), m, 2, +) covering EBS on Z2_M. This can be equivalently written as a 2 (m , m, 2, &) covering EBS on Z2_M, so from Lemma 6.3 there is a 2 2 (2m ,2m,2)BSonZ2_Mrelative to Z2 . Apply Theorem 6.2. For (ii), as noted above there is a (m2, m, 2, &) covering EBS on

Z2_M. Apply Theorem 6.1 to this covering EBS and the BSs of (i). For (iii), apply Theorem 2.4. The case d=0 results from a (m((m&1)Â2), m, 4, +) covering EBS on M, which exists by assumption. Each case d1 results from the covering EBSs of (ii), noting that the group Gd in (iii) contains a subgroup of index 4 and exponent at most 2d. K

A result broadly equivalent to Corollary 6.4 was proved by Jedwab [27] from the viewpoint of perfect binary arrays, via lengthy computation. (A (m((m&1)Â2), m, 4, +) covering EBS on >r Z , where m2=>r s , i=1 si i=1 i implies the existence of a s1_s2_}}}_sr ``binary supplementary quad- ruple,'' which is the initial object for the recursive constructions in [27].) We believe the method presented here to be much clearer. We have chosen in Corollary 6.4 to begin with a (m((m&1)Â2), m,4,+) covering EBS on M, although it is clear from the proof that it would be 2 sufficient to begin with a (m , m, 2, &) covering EBS on Z2_M. (In the 2 running example of this section, a (3, 3, 4, +) covering EBS on Z3 implies 2 the existence of the required (9, 3, 2, &) covering EBS on Z2_Z3 .) The reason is the following composition theorem for (m((m&1)Â2), m,4,+) covering EBSs.

Theorem 6.5. Suppose there exists a (mi ((mi&1)Â2), mi,4,+)covering 2 EBS on a group Mi of odd order mi for i=1, 2. Then there exists a (m1 m2((m1m2&1)Â2), m1m2,4,+)covering EBS on M1_M2 .

Proof. For i=1, 2 let [Ai, Bi, Ci, Di] be a (mi ((mi&1)Â2), mi,4,+) covering EBS on Mi and let the building block containing mi ((mi+1)Â2) elements be Di . Define the following elements of the group ring Z[M1_M2]:

A=M1A2+A1 M2&A1A2+B1B2&A1B2&B1A2 , B=M C +A M &A C +B D &A D &B C , 1 2 1 2 1 2 1 2 1 2 1 2 (6) C=M1 A2+C1 M2&C1A2+D1B2&C1B2&D1A2 , D=M C +C M &C C +D D &C D &D C . 1 2 1 2 1 2 1 2 1 2 1 2 =

File: 582A 279633 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3101 Signs: 2229 . Length: 45 pic 0 pts, 190 mm 46 DAVIS AND JEDWAB

We firstly show that each of these elements can be regarded as a subset of

M1_M2 (so the coefficients in the group ring are 0 or 1). Consider the following subsets of M1_M2 :

S1=(A1 &B1)_(M2"A2),

S2=((M1"A1)&(M1"B1))_A2,

S3=(A1 &(M1"B1))_(M2"B2),

S4=((M1"A1)&B1)_B2.

By inspection the Si have empty pairwise intersection. Let T=A1 & B1 , and note that we can write (M1"A1) & (M1"B1)as(M1"A1)"(B1"T). Then in the group ring we have S1=T(M2&A2), S2=(M1&A1&B1+T)A2,

S3=(A1&T)(M2&B2) and S4=(B1&T)B2, from which S1+S2+ S3+S4=A. Since the Si have empty pairwise intersection, A is therefore a subset of M1_M2 . Similar arguments hold for B, C and D. We claim that [A, B, C, D] is a (m1 m2((m1m2&1)Â2), m1m2,4,+) covering EBS on M1_M2 . To show that A, B, C and D have the correct 2 size, note that for i=1, 2 we have |Mi |=mi ,|Ai|=|Bi|=|Ci|= mi((mi&1)Â2) and |Di|=mi((mi+1)Â2). Then from (6) we have

|A|=|M1A2|+|A1M2|&|A1A2|+|B1B2|&|A1B2|&|B1A2|

=m1m2((m1m2&1)Â2), and similar calculations show that |B|=|C|=m1m2((m1m2&1)Â2) and |D|=m1m2((m1m2+1)Â2). It remains to establish the character properties for [A, B, C, D].

Let / be a nonprincipal character of M1_M2 . Firstly consider the case when / is nonprincipal on M1 and M2 . Then the terms in (6) involving M1 or M2 have a character sum of 0. By the definition of covering EBS, for i=1, 2 exactly one of Ai , Bi , Ci and Di has nonzero character sum (with modulus mi). Since each term X1Y2 , where X, Y # [A, B, C, D], occurs exactly once in (6) it follows that exactly one of A, B, C and D has nonzero character sum (with modulus m1 m2). By symmetry in the subscripts 1 and 2 in the equations (6), it is now sufficient to consider the case when / is principal on M1 and nonprincipal on M2 . Exactly one of the building blocks A2 , B2 , C2 and D2 then has a nonzero character sum. If this building block is A2 then /(B)=/(D)=0, /(C)=(|M1 |&|C1|&|D1|) /(A2)=0 and /(A)=(|M1|&|A1|&|B1|) /(A2)=m1/(A2), which has modulus m1 m2 . If instead the building block with nonzero character sum is B2 , C2 or D2 then similar calculations show that C, B or D respectively has nonzero character sum (with modulus m1 m2) while the rest of A, B, C and D have zero character sum. K

File: 582A 279634 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3215 Signs: 2173 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 47

Theorem 6.5 is based on a construction of Turyn [56] involving incidence matrices of Hadamard difference sets known as Williamson matrices. (The form of the construction given in [56] corresponds to the equations for the Si in the above proof). In Theorem 6.5 we have established additional character properties of Turyn's construction for use in Corollary 6.4. The construction of Hadamard difference sets now relies on finding initial (m((m&1)Â2), m, 4, +) covering EBSs on groups of order m2 (which, from the relationship between covering EBS parameters given after Theorem 2.4, must be odd). The following examples are known.

Theorem 6.6. There exists a (m((m&1)Â2), m,4,+) covering EBS on the following groups M of order m2 : (i) M is the trivial group. 2 (ii) M=Z3: , where :1. 4 (iii) M=Zp , where p is an odd prime. Theorem 6.6(ii) is due to Arasu, Davis, Jedwab and Sehgal [2]. Theorem 6.6(iii) is due to Chen [5], who built on a succession of papers 2 4 4 devoted to finding a Hadamard difference set in Z2_Zp or Z4_Zp . Initially Xia [58] constructed such a difference set for all primes p con- gruent to 3 modulo 4. Xiang and Chen [59] then showed that this construction could be viewed as depending on subsets whose character properties correspond to those of a covering EBS. Next, van Eupen and Tonchev [22] found a difference set example for p=5 and subsequently Wilson and Xiang [57] found an example for p=13 and p=17. Finally Chen [5] solved the problem for all odd primes p by constructing the covering EBS of Theorem 6.6(iii).

Corollary 6.7. Let M be either the trivial group or the group 2 4 > Z :i_> Z , where : 1 and where p is an odd prime, and let i 3 j pj i j |M|=m2. Then the following exist: (i) A (m((m&1)Â2), m,4,+)covering EBS on M. 2d&1 2 d (ii) A (2 m ,2 m,2) BS on Gd_M relative to any subgroup 2d of order 2, where d1 and Gd is any group of order 2 and exponent at most 2d. 2d&1 2 d (iii) A (2 m ,2 m,4,&) covering EBS on Gd_M, where d1 2d d and Gd is any group of order 2 and exponent at most 2 . d (iv) A Hadamard difference set with N=2 minGd_M,where d0 2d+2 d+2 and Gd is any group of order 2 and exponent at most 2 . Proof. For (i), apply Theorem 6.5 to the initial covering EBSs of Theorem 6.6. Then (ii), (iii) and (iv) follow from Corollary 6.4. K

File: 582A 279635 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 2982 Signs: 2272 . Length: 45 pic 0 pts, 190 mm 48 DAVIS AND JEDWAB

Corollary 6.7(iv) is one of the central results of the paper, together with Corollary 5.3 (once again, the groups involved are implicitly abelian). There are no other abelian groups known to contain Hadamard difference sets. The case where M is trivial is due to Kraemer [31]. When combined with an exponent bound given by Turyn [55], Kraemer's result states that Hadamard difference sets exist in abelian groups of order 22d+2 if and only if the exponent is at most 2d+2. For the case where M is nontrivial many nonexistence results depending on number theoretic conditions are known (for details see Davis and Jedwab [13]).

7. FAMILIES OF BUILDING SETS

In this section we demonstrate the full power of the recursive construction for BSs of Section 4 by applying Theorem 4.3 systematically to a small initial set of BSs. This produces several families of BSs, including as special cases all those previously found in Sections 5 and 6 for the purpose of constructing difference sets. The BSs constructed here will be used in Section 8 to deduce the existence of families of semi-regular RDSs, and in Section 9 when we discuss nonabelian groups. Since the construction is based on Theorem 4.3, all the BSs are defined on a group G relative to an elementary abelian subgroup U, although G will not necessarily be a p-group. We are now interested not only in the groups G on which BSs with given parameters exist, but also in the different possible subgroups U. The arguments in this section are probably the most difficult in the paper! In general, we wish to find (a, - at, t) BSs for which the number of building blocks t is large. We have seen in Section 5 that difference sets in several parameter families can be constructed from BSs using Theorem 3.2 only when t is large, and by Lemma 2.1 a (a, - at, t) BS on a group G is clearly a more general object than a (as, - at, tÂs) BS on a group G$ containing G as a subgroup of index s. On the other hand, we have seen in Section 6 that it is possible to trade the growth in t under the recursive application of Theorem 4.3 for a more general form for the group G. For d d d d+1 example, for any d1, by Theorem 5.1(i) there is a (2 ,2,2 )BSonZ2 2d&1 d relative to Z2 whereas by Corollary 6.7(ii) there is a (2 ,2 ,2)BSon any group of order 22d and exponent at most 2d relative to any subgroup of order 2. The first set of BSs is more general in that the number of building blocks is 2d rather than 2, but the second set of BSs is more general in that the group rank can be as small as 2 rather than d+1. (Lemma 2.1 allows some of the second set of BSs to be constructed from the first, but not when the group rank is less than d+1). We now present a general recursive application of Theorem 4.3 which gives the result of applying Lemma 2.1 prior to Theorem 4.3 as desired, throughout the recursion, to

File: 582A 279636 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3329 Signs: 2861 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 49 the BSs with the smallest number of building blocks. As before, the recursion is controlled by the parameter d. We introduce a new parameter c, ranging from a minimum value cÄ up to d, which will be used in applications of Theorem 7.1 to indicate the maximum exponent pc of the Sylow p-subgroup of the group Gd, c .

Theorem 7.1. Let p be prime and let r1 and cÄ 1. For each dcÄ and for each c in the range cÄ cd let Kd, c be a set of ordered pairs (d+c&2cÄ +2)r (Gd, c , Ud, c), where Gd, c is a group of order p a containing a sub- r group Ud, c$Z p . Suppose that for each (GcÄ , cÄ , U cÄ cÄ )#KcÄ , cÄ there exists a r r r ( p a, p - at, p t) BS on GcÄ , cÄ relative to UcÄ , cÄ . Suppose also that, for each d>cÄ and for each c in the range cÄ cd and for each (Gd, c , Ud, c)#Kd,c, 2r Gd,c contains a subgroup Q$Zp and subgroups H0=Ud, c , H1 , ..., Hp r of order pr (corresponding to hyperplanes when viewed as subgroups of Q) such that, for all Hi{H0 ,

(i) Kd&1, c contains an ordered pair isomorphic to (Gd, c ÂHi , QÂHi) for each c in the range cÄ cd&1, and r (ii) Gd, dÂHi contains a subgroup SÂHi of index p such that Kd&1, d&1 contains an ordered pair isomorphic to (SÂHi , QÂHi). Then for each dcÄ and for each c in the range cÄ cd and for each (d+c&2cÄ +1)r (d&cÄ +1)r (d&c+1)r (Gd, c , Ud, c)#Kd,c there exists a ( p a, p -at, p t) BS on Gd, c relative to Ud, c . Proof. The proof is by induction on d. The case d=cÄ is true by assumption. Assume the case d&1 to be true (for each value of c in the range cÄ c d &1) and consider Hi{H0 . For each value of c in the range cÄ c d we can apply Theorem 4.3 with G=Gd, c and H0=Ud, c to establish the case d, provided there exists a ( p(d+c&2cÄ ) r a , p ( d & cÄ ) r - at, p(d&c)rt)

BS on Gd, c ÂHi relative to QÂHi . The subsequent analysis depends on whether cd&1 or c=d. When cd&1, by assumption Kd&1, c contains an ordered pair isomorphic to (Gd, c ÂHi , QÂHi ) so the required BS is given by the inductive hypothesis with the value c. When c=d, there is no inductive hypothesis with the value c. But by assumption Gd, d ÂHi contains a r subgroup SÂHi of index p such that Kd&1, d&1 contains an ordered pair isomorphic to (SÂHi , QÂHi ). Therefore by the inductive hypothesis there (2d&2cÄ &1)r (d&cÄ ) r r exists a ( p a, p - at, p t)BSonSÂHi relative to QÂHi . Then r Lemma 2.1 with s=p gives the required BS on Gd, d ÂHi relative to QÂHi . This completes the induction. K The special case cÄ = c =1 of Theorem 7.1 is similar to Theorem 4.6. We shall consider groups Gd, c in Theorem 7.1 whose Sylow p-subgroup has exponent at most pc. The minimum value of c is cÄ , which will be either

File: 582A 279637 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3452 Signs: 2560 . Length: 45 pic 0 pts, 190 mm 50 DAVIS AND JEDWAB

1 or 2. (For example, we must take cÄ =2 when considering an initial (4, 2 -2, 2) BS on a group G of order 8 relative to a subgroup U of order 2, which we shall later show exists only if U is contained in a sub- (d+c&2cÄ +2)r group of G isomorphic to Z4 .) Then, since Gd, c has order p a, for fixed d>2 we see that for larger values of c the number of building blocks p(d&c+1)rt is smaller but the Sylow p-subgroup is allowed to have smaller rank (and larger exponent). Furthermore by Theorem 2.2 we can construct a semi-regular RDS in groups G$d, c for which the exponent of the Sylow p-subgroup is a multiple of pc+(d&c+1)r. For r>1 this means we can achieve a smaller rank for G$d, c , by taking c large, at the cost of a smaller maximum exponent. For further details see Section 8. Theorem 7.1 is particularly straightforward to apply when pr=2 and we now do so, after giving another construction for a BS from a covering EBS similar to Lemma 6.3.

Lemma 7.2. Let M be a group of odd order m2. Suppose there exists a 2 2 3Â2 (m , m,2,&) covering EBS on Z2_M. Then there exists a (4m ,2 m,2)

BS on Z4_Z2_M relative to the subgroup of order 2 contained within Z4 .

4 2 2 Proof. Let Z4_Z2=(x, y|x =y =1) and let [A, B] be a (m , m,2,&) covering EBS on G=(y)_M. Define the subsets C=(1+x) 2 2 (A+x (G"A)) and D=(1+x)(B+x (G"B)) of Z4_Z2_M. A similar method to the proof of Lemma 6.3 shows that [C, D] is a (4m2,23Â2m,2) BS on (x)_G relative to (x2). K

Corollary 7.3. Let M be either the trivial group or the group 2 4 2 > Z : _> Z , where : 1 and where p is an odd prime, and let |M|=m . i 3 i j p j i j (i) For each d and c satisfying 1cd, there exists a (2d+c&1m2, d d&c+1 2 m,2 ) BS on Gd, c_M relative to any subgroup of order 2, where d+c c Gd, c is any group of order 2 and exponent at most 2 . (ii) For each d and c satisfying 2cd, there exists a (2d+c&2m2, (2d&1)Â2 d&c+1 2 m,2 ) BS on Gd, c_M relative to any subgroup Ud, c of order 2 which is contained within a subgroup of Gd, c isomorphic to Z4 , where d+c&1 c Gd, c is any group of order 2 and exponent at most 2 . Proof. By Corollary 6.7(i) there exists a (m((m&1)Â2), m, 4, +) covering EBS on M. Therefore (as in the proof of Corollary 6.4) there exists a 2 (m , m, 2, &) covering EBS on Z2_M. We shall now apply Theorem 7.1 with pr=2. 2 For (i), let a=m and cÄ = t =1 and take Kd, c to be the set of all ordered d+c pairs (Gd, c_M, Ud, c) for which Gd, c is a group of order 2 and exponent c at most 2 and Ud, c is a subgroup of order 2. By Lemma 6.3 there exists

File: 582A 279638 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3207 Signs: 2441 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 51

2 2 a(2m,2m,2)BSonZ2_Mrelative to Z2 , which satisfies the condition on 2 KcÄ , cÄ . Let Q$Z2 be a subgroup of Gd, c containing H0=Ud, c and let the subgroups of Gd, c of order 2, corresponding to hyperplanes when viewed as subgroups of Q,beH0,H1and H2 . For Hi{H0 and d>1, clearly Gd, c ÂHi has order 2d+c&1 and exponent at most 2c when cd&1, and it is easily shown that Gd, d ÂHi contains a subgroup SÂHi (containing QÂHi)of index 2 and exponent at most 2d&1. The result follows from Theorem 7.1. 2 For (ii), let a=2m , cÄ =2 and t=1 and take Kd, c to be the set of all d+c&1 ordered pairs (Gd, c_M, Ud, c) for which Gd, c is a group of order 2 c and exponent at most 2 and Ud, c is a subgroup of order 2 contained within a subgroup of Gd, c isomorphic to Z4 . By Lemma 7.2 there exists a 2 3Â2 (4m ,2 m,2)BSonZ4_Z2_Mrelative to the subgroup of order 2 contained within Z4 , which satisfies the condition on KcÄ , cÄ . The remainder of the proof is similar to that of (i), with an additional condition on QÂHi which we now demonstrate. By Lemma 4.4, QÂHi is contained in a subgroup of Gd, c ÂHi isomorphic to Z4 for cd. In the case c=d, this implies that the subgroup SÂHi of Gd, d ÂHi can be chosen as above so that QÂHi is contained in a subgroup of SÂHi isomorphic to Z4 when d>2. K

The examples of BSs used at the beginning of this section to introduce Theorem 7.1 are given by the extreme cases c=1 and c=d of Corollary

7.3(i). We now show the condition in Corollary 7.3(ii), that Ud, c is contained within a subgroup of Gd, c isomorphic to Z4 , to be necessary. This is a consequence of the character sum modulus 2(2d&1)Â2m being non- integer.

Lemma 7.4. Suppose there exists a (a, m, t) BS on a group G_W relative to a subgroup U of G, where G is a 2-group and m is not integer. Then rank(GÂU)=rank(G).

Proof. Suppose, for a contradiction, that rank(GÂU)

A RDS in a group U_W relative to U is said to be ``splitting''; the conclusion rank(GÂU)=rank(G) implies in particular that the BS in Lemma 7.4 cannot have this form. We have seen how to apply Theorem 7.1 when pr=2. We now examine conditions (i) and (ii) of the Theorem more closely in order to indicate

File: 582A 279639 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3204 Signs: 2567 . Length: 45 pic 0 pts, 190 mm 52 DAVIS AND JEDWAB how we shall apply it when pr>2. Suppose we begin with the BSs of 2r r Corollary 4.2, taking a=t=1 and K1, 1=[(Zp , Zp)]. Then clearly we can 3r r satisfy condition (i) by choosing K2, 1=[(Zp , Zp)]. We can satisfy condition (ii) by choosing K2, 2 to be the set of all ordered pairs (G2, 2 , U2, 2) for 4r 2 which G2, 2 has order p and exponent at most p subject to the constraint, r for each Hi{H0 , that G2, 2 ÂHi contains a subgroup of index p and exponent p containing QÂHi . We shall show that these constraints on the

Hi{H0 are all implied by the single constraint that G2, 2 ÂU2, 2 contains a subgroup of index pr and exponent p (by suitable choice of the subgroup 2r Q$Zp ). In other words, the single constraint is that G2, 2 ÂU2, 2 attains its 2 2r&2 r+1 maximum exponent p at most r times. For example, if G2, 2=Zp _Zp2 r (where r>1) and we write the subgroup U2, 2$Zp as being contained within r direct factors of G2, 2 then all choices of U2, 2 are allowed, except 2r&2 possibly U2, 2 being contained within the subgroup Zp . This demonstrates that even when all positions of UcÄ , cÄ within GcÄ , cÄ are allowed, not all positions of Ud, c within Gd, c are necessarily allowed. Continuing to the next level d=3, we can similarly take K3, 1= 4r r [(Zp , Zp)]. We then satisfy condition (i) by choosing K3, 2 to be the set of 5r all ordered pairs (G3, 2 , U3, 2) for which G3, 2 has order p and exponent at 2 most p , provided that (G3, 2 ÂHi , QÂHi ) is contained in K2, 2 . This requires r (G3, 2 ÂHi)Â(QÂHi ) to contain a subgroup of index p and exponent p for each Hi{H0 . We shall show that these constraints are all implied by the 2r single constraint that G3, 2 ÂU3, 2 contains a subgroup of index p and exponent p. Thus, a constraint for (d, c)=(2, 2) forces a constraint for (d, c)=(3, 2) and will likewise propagate to all (d, 2) with d>2. We then satisfy condition (ii) by choosing K3, 3 to be the set of all ordered pairs 6r 3 (G3, 3 , U3, 3) for which G3, 3 has order p and exponent at most p , subject to two constraints. The first, that G3, 3 ÂU3, 3 contains a subgroup of index pr and exponent at most p2, arises directly from condition (ii) as a result 2 3 of the increase in exponent from p to p . The second, that G3, 3ÂU3, 3 contains a subgroup of index p3r and exponent p, is inherited via condition (ii) 2r r from the constraint on K2, 2 . Note that K2, 2 contains (Zp 2 , Z p) and K3, 3 2r r contains (Zp 3 , Zp). This illustrates that the rank of the Sylow p-subgroup of Gd, c need not increase from the minimum value of 2r, as required so that 2r the subgroup Q$Zp exists. Following the pattern of the above example, we now produce an explicit form for allowable ordered pairs (Gd, c , Ud, c) from Theorem 7.1, involving j the existence of a subgroup of Gd, c ÂUd, c of exponent p for each j in the range cÄ j c &1. Although the following theorem constructs many such ordered pairs, it is necessary to check only that the subgroup conditions hold for a particular ordered pair (Gd, c , Ud, c) and a particular value of d and c to conclude that the stated BS exists for this ordered pair. For the moment we take Gd, c to be a p-group.

File: 582A 279640 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3771 Signs: 3066 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 53

Theorem 7.5. Let p be prime and let r1. Suppose there exists a r r r 2r ( p a, p - at, p t) BS on any p-group GcÄ , cÄ of order p a and exponent at most cÄ r cÄ (2r&1)&2r&1 p relative to any subgroup UcÄ , cÄ $ Z p , where a>p and cÄ 1. Then for each d and c satisfying cÄ cd, there exists a ( p(d+c&2cÄ +1)ra, (d&cÄ +1)r (d&c+1)r (d+c&2cÄ +2)r p -at, p t) BS on any p-group Gd, c of order p a c r and exponent at most p relative to any subgroup Ud, c$Zp, where, for each j in the range cÄ j c&1, Gd, cÂUd, c contains a subgroup of index p(d+c&2j&1)r and exponent at most p j.

Proof. We shall apply Theorem 7.1, taking Kd, c to be the set of all ordered pairs (Gd, c , Ud, c) for which Gd, c is any p-group of order (d+c&2cÄ +2)r c p a and exponent at most p and Ud, c is any subgroup isomorphic r to Zp provided that, for each j in the range cÄ j c &1, Gd, cÂUd, c contains a subgroup of index p(d+c&2j&1)r and exponent at most p j. The condition on KcÄ , cÄ is true by assumption. Consider d and c satisfying d>cÄ and (d+c&2cÄ +2)r c cÄ c d . Since Gd, c has order p a and exponent at most p , and by assumption a>pcÄ (2r&1)&2r&1, it is straightforward to show that 2r rank(Gd, c)>2r&1. Therefore Gd, c contains a subgroup Qd, c$Z p containing Ud, c . Choose the subgroups Hi of Gd, c corresponding to hyperplanes of Qd, c so that H0=Ud, c , and consider Hi{H0 . The result follows from Theorem 7.1 subject to the following two conditions. Firstly, when (d+c&2j&2)r cd&1, (Gd, cÂHi )Â(Qd, c ÂHi ) contains a subgroup of index p and exponent at most p j for each j in the range cÄ j c &1. Secondly, r Gd, d ÂHi contains a subgroup SÂHi (containing Qd, d ÂHi ) of index p and exponent at most pd&1 such that, for each j in the range cÄ j d &2, (2d&2j&3)r (SÂHi )Â(Qd, d ÂHi ) contains a subgroup of index p and exponent at most p j. We now demonstrate the existence of the required subgroups when cd&1 and when c=d to complete the proof. For cd&1 or c=d, let

r w

Gd, c= ` Zp1+:u_ ` Zp1+;u ,(7) u=1 u=1 where wr, Qd, c is contained in the first 2r direct factors of Gd, c , Ud, c is contained in the first r direct factors of Gd, c , c&1:u0 for each u, and c&1;1;2}}};w0. By the third isomorphism theorem for groups, (Gd, c ÂHi )Â(Qd, c ÂHi )$Gd, cÂQd, c and so

r r w

(Gd, c ÂHi )Â(Qd, c ÂHi )$ ` Zp:u_ ` Zp;u_ ` Zp1+;u.(8) u=1 u=1 u=r+1

File: 582A 279641 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3493 Signs: 2474 . Length: 45 pic 0 pts, 190 mm 54 DAVIS AND JEDWAB

Furthermore from (7),

r w

Gd, c ÂUd, c$ ` Zp:u_ ` Zp1+;u ,(9) u=1 u=1 and by assumption (9) contains a subgroup T of index p(d+c&2j&1)r and exponent at most p j for each j in the range cÄ j c &1. Let s=s( j ) be the largest u for which ;uj.Ifsrthen it clearly follows that (8) contains a subgroup (isomorphic to T ) of index p(d+c&2j&1)rÂpr and exponent at j most p .Ifss and so (8) contains a subgroup of exponent at most p j and index at most p(c&1&j)r+(c&1&j)s< p(c&1&j)r+(d&1&j)r (since cd). Therefore for sr or s

Finally, for cÄ 2, let Ud, c be additionally constrained to be contained r within a subgroup of Gd, c isomorphic to Zp 2 for each dcÄ and for each c in the range cÄ c d . The above induction step for d>cÄ can be modified as follows. The required BS on Gd, c ÂHi relative to Qd, c ÂHi exists provided

File: 582A 279642 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3393 Signs: 2744 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 55

r Qd, c ÂHi is contained in a subgroup of Gd, c ÂHi isomorphic to Z p 2 when cd&1, and provided Qd, d ÂHi is contained in a subgroup of r SÂHi isomorphic to Zp2 . Apply Lemma 4.4 with G=Gd, c , H0=Ud, c and Q=Qd, c to show that Qd, c ÂHi is contained in a subgroup of Gd, c ÂHi r isomorphic to Zp2 for cd. This establishes the result when cd&1. It remains to show that if Q$=Qd, dÂHi is contained in a subgroup of r G$=Gd, dÂHi isomorphic to Zp 2 (as just demonstrated) then we can choose S$=SÂHi consistently with the previous procedure so that Q$ is also con- r tained in a subgroup of S$ isomorphic to Z p 2 . In other words, given that the r direct factors of G$ÂQ$ corresponding to the position of Q$inG$ each attains an exponent of at least p, we must choose the subgroup S$ÂQ$ consistently so that none of these direct factors is removed. The reduction in exponent to pd&2 can proceed as before since d&2cÄ &11, and the choice of S$ÂQ$ with minimal exponent does not require the removal of any of the r direct factors unless |G$ÂQ$|p2r&1. It is straightforward to show that this inequality is false and so the induction proof carries over. K

Theorem 7.5 gives conditions on subgroups of Gd, c ÂUd, c for each c in the range cÄ c d and for each j in the range cÄ j c &1. We shall show in the following four corollaries that in particular cases some of these conditions are implied by others while some are guaranteed to hold because of the order and exponent restrictions on Gd, c . Each of the corollaries is based on one of the following four sources, to which we apply Theorem 4.3 if necessary to obtain initial BSs comprising prt building blocks on any p-group GcÄ , cÄ of fixed order and bounded exponent relative to any sub- r group UcÄ , cÄ $ Z p .

Theorem 7.6. For each r1, the following exist:

r r r 2r r (i) A (p , p , p ) BS on Zp relative to Zp , where p is prime and r1. r rÂ2 2r r (ii) A (p , p ,1)BS on Z p relative to Zp , where p is an odd prime and r1. r rÂ2 r r (iii) A (2 ,2 ,1)BS on Z 4 relative to Z2 , where r1. 2 2 2 (iv) A (8, 4, 2) BS on Z4_Z2 relative to the subgroup Z2 of Z4 .

Theorem 7.6(i) is just a restatement of Corollary 4.2. Theorem 7.6(ii) and (iii) are equivalent to Jungnickel's result [28] that semi-regular RDSs, with parameters ( pr, pr, pr, 1) and (2r,2r,2r, 1) respectively, exist for the stated groups and subgroups. (Nonexistence results for RDSs show that no other abelian group and subgroup can be substituted in Theorem 7.6(ii) for r=1 [26] or r=2 [36], or in Theorem 7.6(iii) for any r [24].) Theorem 7.6(iv) is due to Arasu and Sehgal [3], as described in Section 2.

File: 582A 279643 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3298 Signs: 2588 . Length: 45 pic 0 pts, 190 mm 56 DAVIS AND JEDWAB

Suppose we apply Theorem 7.5 directly to the BSs of Theorem 7.6(i) to obtain BSs with parameters ( p(d+c&1)r, pdr, p(d&c+1)r). By Theorem 2.2 this gives semi-regular RDSs with parameters ( p2dr, pr, p2dr, p(2d&1)r) for d1. For fixed pr, successive values of the first RDS parameter p2dr differ by a factor p2r. We show in the first corollary that we can reduce this factor to p2 by producing BSs which give RDSs with parameters ( p2dr+2i, pr, p2dr+2i, p(2d&1)r+2i) for each i in the range 0i

Corollary 7.7. Let p be prime and let i and r satisfy 0i1 and c=d, Gd, c ÂUd, c contains a subgroup of index pr and exponent at most pd&1 and where, for i>0 and c in the range max[1, ((d&1) r+i)Â(r+i)]

r+i r+i r+i 2(r+i) Proof. We begin with a ( p , p , p )BSonZp relative to r+i i Zp from Theorem 7.6(i), and use Lemma 4.5 with W=Z p to obtain a r+i r+i r+i 2r+i r ( p , p , p )BSonZp relative to Zp . This provides the initial BS i on GcÄ , cÄ relative to UcÄ , cÄ in Theorem 7.5, taking a=t=p and cÄ =1. We then obtain the required BS on Gd, c relative to Ud, c provided that, for each c in the range 1cd, the following condition on j is satisfied for each j in the (d+c&2j&1)r range 1 j c&1: Gd, cÂUd, c contains a subgroup of index p and exponent at most p j. We now show that this set of conditions can be replaced by the smaller set stated in the theorem by distinguishing four cases: firstly c((d&1) r+i)Â(r+i), when no condition on j will be needed; secondly cd&1, when the condition on j=1 will suffice; thirdly c=d and i=0, when the condition on j=d&1 will suffice; and fourthly c=d and i>0, when the condition on j=1 and j=d&1 will together suffice. (In the theorem, the condition on j=1 is written in the equivalent form: rank(Gd, c ÂUd, c)2r+i.) Since the range of j is 1 j c&1, we shall assume c>1 throughout. (d+c&1) r+i c The group Gd, c ÂUd, c has order p and exponent at most p ,so c :u we can write it as >u=1 Zpu , where :u0 and

: u:u=(d+c&1) r+i. (10) u=1

File: 582A 279644 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3102 Signs: 2487 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 57

wj Clearly Gd, c ÂUd, c contains a subgroup of index p and exponent at most pj, where

wj= : (u&j) :u. (11) u=j+1

We shall repeatedly use the fact that

C C&j C : (u&j) :u : (u&;) :u (12) u=j+1 \C&;+ u=j+1 for any integer ;j. In the first case, when c((d&1) r+i)Â(r+i), we consider j in the range 1 j c&1. Put ;=0 and C=c in (12) and substitute from c (11) to show that wj((c&j)Âc) u=j+1 u:u . (10) then implies that wj ((c&j)Âc)((d+c&1) r+i). Rearrangement gives wj&(d+c&2j&1)r ( j(r+i)Âc)(c&((d&1) r+i)Â(r+i))&i( j&1). Since c((d&1) r+i)Â

(r+i), we obtain wj&(d+c&2j&1)r0. Therefore by the definition (d+c&2j&1)r of wj , Gd, c ÂUd, c always contains a subgroup of index p and exponent at most p j for each j in the range 1 j c&1. In the second case, when cd&1, we assume the condition on j=1 holds and consider j in the range 2 j c&1. Take ;=1 and C=c in (12) and use (11) to deduce that wj((c&j)Â(c&1)) w1. Now the condition on j=1 can be written as w1(d+c&3)r and so wj&(d+c&2j&1)r (( j&1)Â(c&1))(c+1&d)r. Since c d & 1, we obtain wj &(d+c& 2j&1)r0. In the third case, when c=d and i=0, we assume the condition on j=d&1 holds and consider j in the range 1jd&2. We can rewrite (11) d&1 as wj=(d&j) :d+u=j+1 (u&j) :u and then put ;=0 and C=d&1 in d&1 (12) to show that wj(d&j) :d+((d&1&j)Â(d&1)) u=j+1 u:u . (10) then implies that wj(d&j) :d+((d&1&j)Â(d&1))((2d&1) r&d:d ).

Hence wj&(2d&2j&1)r( jÂ(d&1))(:d&r). Now the condition on j=d&1 gives wd&1r, which from (11) is equivalent to :dr. Therefore wj&(2d&2j&1)r0. In the fourth case, when c=d and i>0, we assume the condition on j=1 and j=d&1 to hold and consider j in the range 2 j d&2. Rewrite d&1 (11) as wj=(d&j) :d+u=j+1 (u&j) :u. Take ;=1 and C=d&1 in (12) and use (11) to deduce that wj(d&j) :d+((d&1&j)Â(d&2)) (w1&(d&1) :d ). Now the condition on j=1 gives w1(2d&3)r and so wj&(2d&2j&1)r(( j&1)Â(d&2))(:d&r). The condition on j=d&1 then gives wd&1=:dr, and so wj&(2d&2j&1) r0. This completes the proof. K

File: 582A 279645 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 2708 Signs: 1957 . Length: 45 pic 0 pts, 190 mm 58 DAVIS AND JEDWAB

The condition on Gd, d ÂUd, d in Corollary 7.7 is a consequence of the increase in group exponent from pd&1 to pd. The condition on 2r+i rank(Gd, c ÂUd, c) derives from the initial BS, which is defined on Zp .We note that the range of values of d, i and c in Corollary 7.7 for which

Gd, c ÂUd, c must be constrained could be slightly improved because it is sufficient to ensure that wj&(d+c&2j&1) r<1 in the proof rather than wj&(d+c&2j&1) r0. We have chosen the presented form for clarity; it is straightforward to check in individual cases whether the conditions are guaranteed to hold.

Corollary 7.8. Let p be an odd prime and let i and r satisfy 0i<2r. r+i (r+i)Â2 2r+i r There exists a ( p , p ,1)BS on Zp relative to Zp . For each d and c satisfying 1cd, there exists a ( p(d+c) r+i, p((2d+1) r+i)Â2, p(d&c+1)r) BS (d+c+1 )r+i c on any group Gd, c of order p and exponent at most p relative to r any subgroup Ud, c$Z p , where, for d>1 and c=d, Gd, c ÂUd, c contains a subgroup of index pr and exponent at most pd&1 and where, for c in the range max[1, (dr+i)Â(2r+i)]

Proof. By Theorem 7.6(ii) and Lemma 4.5 there exists ( pr+i, p(r+i)Â2,1) 2r+i r BS on Zp relative to Zp , as required. Then by Theorem 4.3 there exists 2r+i (3r+i)Â2 r 3r+i r a(p ,p , p )BSonZp relative to Zp . This provides the initial r+i BS on GcÄ , cÄ relative to UcÄ , cÄ in Theorem 7.5, taking a=p and t=cÄ =1. We then obtain the required BS on Gd, c relative to Ud, c provided that, for each c in the range 1cd, the following condition on j is satisfied for each j in the range 1 j c&1: Gd, cÂUd, c contains a subgroup of index p(d+c&2j&1)r and exponent at most p j. These conditions on j are identical to those in the proof of Corollary 7.7 and can likewise be replaced by a smaller set of conditions. The only difference is that here Gd, c ÂUd, c has order p(d+c&1)r+(r+i) rather than p(d+c&1)r+i. Since the replacement of conditions on j in the proof of Corollary 7.7 does not rely on the inequality i

Corollary 7.9. Let i and r satisfy 0i<2r. There exists a (2r+i, (r+i)Â2 r i r r 2 ,1)BS on Z4_Z2 relative to the subgroup Z2 of Z4 . For each d and c satisfying 2cd, there exists a (2(d+c&2)r+i,2((2d&1)r+i)Â2,2(d&c+1)r) (d+c&1) r+i c BS on any group Gd, c of order 2 and exponent at most 2 relative r to any subgroup Ud, c$Z2 , where Ud, c is contained in a subgroup of Gd, c r isomorphic to Z4 and where all of the following hold:

min[r, i] (i) For c=d, Gd, c ÂUd, c contains a subgroup of index 2 and exponent at most 2d&1.

File: 582A 279646 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3277 Signs: 2595 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 59

(ii) For i2 and c=d&1, Gd, cÂUd, c contains a subgroup of index 2r+i and exponent at most 2d&2. (iii) For i>r and c in the range max[1, ((d&2) r+i)Âi]

Corollary 7.7 with p=2, the only difference being that here Gd, c ÂUd, c has order 2(d+c&1) r+(i&r) for i in the range ri<2r rather than 2(d+c&1) r+i for i in the range 0i((d&2)r+i)Âr (which implies that c=d&1 or c=d), we assume the condition on j= c&1 holds and consider j in the range 1 j c&2. Following the same c&1 argument as previously, wj(c&j) :c+((c&1&j)Â(c&1)) u=j+1 u:u (c&j ) :c+((c&1&j)Â(c&1))((d+c&2) r+i&c:c). In the case c=d&1 we obtain wj &(2d&2j&3)r&i(jÂ(d &2))(:d&1 &(r+i)) and the assumed condition on j=d&2 implies :d&1r+i, whereas in the case

File: 582A 279647 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3445 Signs: 2771 . Length: 45 pic 0 pts, 190 mm 60 DAVIS AND JEDWAB

c=d we obtain wj&(2d&2j&2) r&i(jÂ(d&1))(:d&i) and the assumed condition on j=d&1 implies :di. K

r The condition in Corollary 7.9 that Ud, c$Z2 is contained in a subgroup r of Gd, c isomorphic to Z4 is necessary when r+i is odd, by Lemma 7.4.

2 Corollary 7.10. There exists a (8, 4, 2) BS on Z4_Z2 relative to the 2 2 subgroup Z2 of Z4 . For each d and c satisfying 2cd, there exists a 2d+2c&3 2d 2d&2c+3 2d+2c&1 (2 ,2 ,2 ) BS on any group Gd, c of order 2 and expo- c 2 nent at most 2 relative to any subgroup Ud, c$Z2 , where Ud, c is contained 2 in a subgroup of Gd, c isomorphic to Z4 and where, for d>2 and c=d, d&1 Gd, c ÂUd, c contains a subgroup of index 4 and exponent at most 2 . Proof. The (8, 4, 2) BS is given in Theorem 7.6(iv). By Theorem 4.3 and Lemma 4.4 there exists a (32, 16, 8) BS on any group G of order 128 2 and exponent 4 relative to U$Z2 , where U is contained within a subgroup 2 of G isomorphic to Z4 . This provides the initial BS on GcÄ , cÄ relative to UcÄ , cÄ in Theorem 7.5, taking p=r=2, a=8 and t=cÄ =2. By making use of the additional constraint that UcÄ , cÄ is contained within a subgroup isomorphic r to Zp 2 we then obtain the required BS on Gd, c relative to Ud, c provided that, for each c in the range 2cd, the following condition on j is satisfied for each j in the range 2 j c&1: Gd, cÂUd, c contains a subgroup of index 22(d+c&2j&1) and exponent at most 2 j. The remainder of the proof is similar to that of the first and third cases in the proof of Corollary 7.7. As in the proof of Corollary 7.9, the constraint that Ud, c is contained within a sub- 2 group of Gd, c isomorphic to Z4 does not affect the analysis of Gd, c ÂUd, c . When cd&1 we do not assume any condition on j and consider j in the range 2 j c&1. By similar reasoning to that used previously we show that wj&2(d+c&2j&1)(2jÂc)(c+1&d)+( j&c)Âc0. When c=d we assume the condition on j=d&1 to hold, so that :d2, and consider j in the range 2 j d&2. We obtain wj&2(2d&2j&1)

(1Â(d&1))( j(:d&2)+j+1&d)0. K Further examples of BSs can be constructed from those in Corollaries 7.77.10 using Lemma 2.1. We believe the set of BSs so produced to be complete in the sense that no other examples could be obtained from the four sources of initial BSs of Theorem 7.6 using the underlying construction of Theorem 4.3. In Section 8 we discuss further some implications of Corollaries 7.77.10. A fifth source of initial BSs is given by Chen, Ray-Chaudhuri and Xiang's construction [7] of a (22r&1,2r,22r&1,2r&1) semi-regular RDS in 3r&1 r any group G of order 2 and exponent 4 relative to U$Z 2 , where U is r contained within a subgroup of G isomorphic to Z4 and r1 is odd. We can regard this as a (22r&1,2(2r&1)Â2, 1) BS on the stated group and

File: 582A 279648 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3347 Signs: 2714 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 61 subgroup and use the above procedure to obtain an additional corollary. It can be shown that the resulting BSs have parameters in the same families as those of Corollary 7.9 and that (apart from initial examples) the BSs are defined on substantially the same set of groups. Nonetheless this method allows a relaxation of condition (iii) of Corollary 7.9, involving rank(Gd, c ÂUd, c) for i>r, for certain values of i when r5. For example, the minimum rank of Gd, c when r=5 and i=8 can be reduced from 13 to 12 in this way. We conclude this section by modifying Theorem 7.5 to deal with groups whose order is not a prime power.

Theorem 7.11. Let W be a group of order w1, let p be a prime not dividing w and let r1. Suppose there exists a ( praw, pr - awt, prt) BS on 2r any group GcÄ , cÄ _ W , whose Sylow p-subgroup GcÄ , cÄ has order p a and expo- cÄ r cÄ (2r&1)&2r&1 nent at most p , relative to any subgroup UcÄ , cÄ $ Z p , where a>p and cÄ 1. Then for each d and c satisfying cÄ cd, there exists a (d+c&2cÄ +1)r (d&cÄ +1)r (d&c+1)r ( p aw, p -awt, p t) BS on any group Gd, c_W, (d+c&2cÄ +2)r whose Sylow p-subgroup Gd, c has order p a and exponent at most c r p , relative to any subgroup Ud, c$Zp , provided that, for each j in the range (d+c&2j&1)r cÄ j c &1, Gd, cÂUd, c contains a subgroup of index p and exponent at most p j.

Furthermore the theorem remains true for cÄ 2 if UcÄ , cÄ is additionally con- r strained to be contained within a subgroup of GcÄ , cÄ isomorphic to Zp 2 , provided that each Ud, c is likewise contained within a subgroup of Gd, c r isomorphic to Zp 2 . Theorem 7.11 can be proved from Theorem 7.1 in a similar manner to Theorem 7.5: the important calculations involve only the Sylow p-subgroup

Gd, c and not the group W. We did not give this more general form earlier in order to avoid the introduction of several new parameters at the same time. In particular cases, the conditions on subgroups of Gd, c ÂUd, c in Theorem 7.11 can be replaced by a smaller set of conditions, as in Corollaries 7.77.10. Theorem 7.11 could have been used in this way to prove the results of Corollary 7.3. Moreover Theorem 7.11 has potential use in determining the existence of new families of BSs, subject to finding appropriate initial BSs on groups whose order is not a prime power.

8. APPLICATION TO SEMI-REGULAR RELATIVE DIFFERENCE SETS

In this section we use Theorem 2.2 to deduce the existence of families of r semi-regular RDSs in groups G relative to subgroups U$Zp from the BSs

File: 582A 279649 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3053 Signs: 2472 . Length: 45 pic 0 pts, 190 mm 62 DAVIS AND JEDWAB constructed in Section 7. In particular we show that the order of G can grow without bound for fixed rank 2r. We are not aware of any abelian groups G known to contain semi-regular RDSs relative to an elementary abelian subgroup which are not covered by these results. We begin with the special case of a subgroup U of order pr=2. This is a rich source of RDSs in groups G whose order has distinct prime factors, deriving from the (m((m&1)Â2), m, 4, +) covering EBSs on groups M constructed in Corollary 6.7(i).

Corollary 8.1. Let M be either the trivial group or the group 2 4 > Z :i_> Z , where : 1 and where p is an odd prime, and let i 3 j pj i j |M|=m2. For each d1 the following exist:

2d 2 2d 2 2d&1 2 (i) A (2 m ,2,2 m,2 m ) semi-regular RDS in Gd_M relative 2d+1 to any subgroup Ud of order 2, where Gd is any group of order 2 and exponent at most 2d+1. 2d+1 2 2d+1 2 2d 2 (ii) A (2 m ,2,2 m ,2 m)semi-regular RDS in Gd_M relative to any subgroup Ud of order 2 contained in a subgroup of Gd 2d+2 isomorphic to Z4 , where Gd is any group of order 2 and exponent at most 2d+2.

Proof. For (i), Gd contains a subgroup of index 2 and exponent at most d 2 containing Ud . Apply Theorem 2.2 to the case c=d of Corollary 7.3(i). d+1 For (ii), Gd contains a subgroup S of index 2 and exponent at most 2 such that Ud is contained in a subgroup of S isomorphic to Z4 . Put c=d in Corollary 7.3(ii), replace d by d+1 throughout and then apply Theorem 2.2. K Corollary 8.1 was proved for trivial M by Ma and Schmidt [37] and for all appropriate groups M (as described in Section 6) by Jedwab [27] via lengthy computations on binary arrays. (Jungnickel [28] earlier noted d&1 that a Hadamard difference set with parameter N=2 m in a group Wd of order 22dm2 can be used to construct a RDS with the parameters of Corollary 8.1(i) in the subset of groups having the ``splitting'' form

Ud_Wd .) Davis [11] used techniques introduced by Turyn [55] to show d+1 that the exponent bound of 2 on Gd in Corollary 8.1(i) is necessary as well as sufficient for trivial M, and Pott [47] proved the corresponding result for the exponent bound 2d+2 in Corollary 8.1(ii). Exponent bounds on Gd when M is nontrivial can be obtained for both parts of the Corollary by similar methods, subject to number theoretic conditions. As already noted, the condition in Corollary 8.1(ii), that Ud is contained in a subgroup of Gd isomorphic to Z4 , is necessary by Lemma 7.4. Jungnickel [28] has shown that the existence of a (2m,2,2m,m) semi-regular RDS (not necessarily in an abelian group) implies the existence of a Hadamard

File: 582A 279650 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3146 Signs: 2577 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 63 matrix of order 2m, which in turn implies that m=1 or m is even (see Seberry and Yamada [51] for a recent survey of Hadamard matrices). Therefore we cannot substitute the value d=0 in Corollary 8.1(ii) when M is nontrivial (although we can when M is trivial). In the remainder of this section, the groups G containing RDSs will be p-groups. The most extensive previous results for RDSs are for r=1, so we next take the order of U to be an odd prime p.

Corollary 8.2. Let p be an odd prime.

(i) For each d1, there exists a ( p2d, p, p2d, p2d&1) semi-regular RDS in any group of order p2d+1 and exponent at most pd+1 relative to any subgroup Ud of order p. (ii) For each d0, there exists a ( p2d+1, p, p2d+1, p2d) semi-regular 2d+2 d+1 RDS in any group Gd of order p and exponent at most p relative to 2 any subgroup Ud of order p, except possibly when G1$Zp2 or when Gd ÂUd$Zp d+1_Zpd for d>1. Proof. The proof is by application of Theorem 2.2 to the following BSs. d For (i), Gd contains a subgroup of index p and exponent at most p containing Ud . The case r=1, i=0, c=d of Corollary 7.7 shows that there exists a ( p2d&1, pd, p) BS on any group of order p2d and exponent at most pd relative to any subgroup of order p. For (ii), put r=1 and i=0 and consider the initial BS of Corollary 7.8 2r+i on Zp together with the case d=1 of Corollary 7.8. This gives a 1Â2 2 2 3Â2 3 ( p, p ,1)BSonZp relative to Zp and a ( p , p , p)BSonZp relative to Zp . This gives the result for d=0 and d=1. For d>1, we have excluded 2 the cases Gd=Zp d+1 and Gd=Ud_Zp d+1_Zpd . Therefore Gd contains a d subgroup S (containing Ud) of index p and exponent at most p for which 2 S$% Ud_Zp d . The case r=1, i=0, c=d of Corollary 7.8 shows that there 2d (2d+1)Â2 2d+1 exists a ( p , p , p) BS on any group Sd of order p and exponent d at most p relative to any subgroup Ud of order p except possibly when 2 2 d>1 and Sd ÂUd$Zp d , which is equivalent to Sd$Ud_Zp d . K Corollary 8.2(i) and many of the cases of Corollary 8.2(ii) were proved by Ma and Schmidt [37]. Davis [11] showed that the exponent bound of pd+1 in Corollary 8.2(i) is necessary, and Ma and Pott [36] established the corresponding bound for Corollary 8.2(ii). It follows from this and from our earlier discussion of the case pr=2 that for p prime, the only abelian groups G of order pw+1 in which the existence of a ( pw, p, pw, pw&1) RDS relative to a subgroup U of order p remains unknown have p odd, 2 w=2d+1, and either G=Zp d+1 or G=U_Zpd+1_Zpd . We have chosen to express the existence condition in Corollary 8.2(ii) for d>1 in terms

File: 582A 279651 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3208 Signs: 2567 . Length: 45 pic 0 pts, 190 mm 64 DAVIS AND JEDWAB

of Gd ÂUd in order to emphasise the importance of the position of Ud within Gd . For r=1, all the RDSs arising from the BS families of Section 7 via Theorem 2.2 can be obtained by taking c=d. However for r>1, each value of c gives rise to different RDSs. For example, take d=4, r=2 and i=0 in Corollary 7.9. Using Theorem 2.2 we obtain a (218,4,218,216) semi- 8+2c regular RDS in certain groups Gc containing a subgroup Sc of order 2 c 2 and exponent at most 2 relative to a subgroup Uc$Z2 . In the case c=2, 10 the maximum exponent of Gc is 2 and the minimum rank is 6, both of 5 which are attained by G2=Z2 10_Z4 and any U2 . In the case c=3, the 9 maximum exponent of Gc is reduced to 2 but the minimum rank is now 3 5, attained by G3=Z2 9_Z8_Z4 and any U3 . In the case c=4, the 8 maximum exponent of Gc is further reduced to 2 but the minimum rank 3 becomes 4, attained by G4=Z2 8_Z16 and any U4 . This shows that for the group G containing the RDS, a small rank is associated with a small exponent. But for the subgroup S on which the underlying BS is defined, we have the usual correspondence between small rank and large exponent. (In the above example, minimum rank 6 and maximum exponent 4, minimum rank 5 and maximum exponent 8, and minimum rank 4 and maximum exponent 16.) For this reason we believe that the natural place to consider exponent bounds is the BS group S rather than the RDS group G. In our opinion the most interesting RDS examples are those for which the underlying BS group has small rank and large exponent. We shall therefore mostly concentrate on the RDSs arising from Corollaries 7.77.10 for the value c=d. Further RDSs can be obtained for c1. We now review the previous state of knowledge for a ( pw, pr, pw, pw&r) semi-regular RDS in an abelian group r G relative to U$Zp , where p is prime and wr>1. The best known nonexistence results are that if such RDSs exist then the following exponent bounds apply: exp(G)p:+1 for w=2: [11]; exp(G)p:+1 for p odd and w=2:+1 [36]; and exp(G)2:+2 for p=2 and w=2:+1 [47]. The r+:&1 first exponent bound is attained for :r and any p by G=Zp :+1_Zp r and any U$Z p . To show this, apply Theorem 2.2 to the BSs of Corol- lary 4.2 followed by Lemma 4.5. The first bound is also attained for :1 2 and p=r=2 by G=Z2:+1_Z4_G:, where U$Z2 is contained in the first :&1 two direct factors of G and where G: is any group of order 2 and exponent at most 4, except possibly G3=Z4 . To show this for :2, apply Theorem 2.2 to the BSs of Theorem 5.1(ii) and (iii). (These examples can also be obtained by taking the minimum value of c in certain of the

File: 582A 279652 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3320 Signs: 2834 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 65

Corollaries of Section 7.) As already discussed, we believe that the RDS group is not the natural place to determine an exponent bound. On the existence side, the previous state of knowledge for r>1 is sum- marised in Table 1. In stating these results we have applied the method of contraction of RDSs (the case t=1 of Lemma 4.5) as appropriate. The construction of Leung and Ma [33] applies to many groups, of which we have included only those of rank less than 5r. This is the only previous construction for which the rank of G does not necessarily grow with the order of the group. The RDSs in Table I can also be combined using the ``product'' construction, due to Davis [8] and Pott [48]. The product construction carries over to BSs and we now state it in this form without proof, but we shall only require the RDS part, namely the case t=t$=1. The principal disadvantage of the product construction is that under repeated application the rank of G is forced to grow.

Theorem 8.3. Let G be a group of order uaa$ containing a subgroup U of order u, and let H and H$ be subgroups of G of order ua and ua$ respectively, where H & H$=U. Suppose there exists a (a, - at, t) BS on H relative to U and there exists a (a$, - a$t$, t$) BS on H$ relative to U. Then there exists a (aa$, - aa$tt$, tt$) BS on G relative to U. (Leung and Ma [33] and Chen, Ray-Chaudhuri and Xiang [7] have also given constructions for semi-regular RDSs in certain p-groups relative to an arbitrary subgroup of order pr, and Schmidt [49] has exhibited a 2 (16, 4, 16, 4) RDS in U_Z4_Z2 with U$Z4 .) To our knowledge no

TABLE I

Source w Min rank(G) Conditions

(A) Jungnickel [28] :: p=2 (V) r+:podd (B) Davis [10] 2:r+::r (C) Davis and Sehgal [16] 2::+1 p=2, r=2, :2 (V) :+2 p=2, r=3, :5 (V) (D) Pott [47] (2d+1): (d+1):p=2, :r, d1 (V) r+(d+1):podd, :>r 2: (E) Leung and Ma [33] 2d: r+2::r,d1, G=U_Z pd (F) Chen, Ray-Chaudhuri 2:&1 (3:&1)Â2 p=2, :r, : odd, and Xiang [7] exp(G)=4 (V)

Note.A(pw,pr,pw,pw&r) semi-regular RDS exists in any abelian group G relative to any r subgroup U isomorphic to Zp provided G has the specified minimum rank and wr1 and the stated conditions are satisfied, where p is prime. (V) indicates the additional condition that r G contains a subgroup isomorphic to Z4 containing U.

File: 582A 279653 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3203 Signs: 2267 . Length: 45 pic 0 pts, 190 mm 66 DAVIS AND JEDWAB abelian groups other than those described have been previously shown to contain semi-regular RDSs. r We now show how these results can be improved for U$Zp by applying Theorem 2.2 to the Corollaries of Section 7, firstly taking p=2.

Corollary 8.4. There exists a (22dr+j,2r,22dr+j,2(2d&1)r+j ) semi- (2d+1)r+j regular RDS in the following groups Gd of order 2 relative to any r subgroup Ud$Z 2 , where j and r satisfy 0j<2r:

(i) When j=0 and r=2. For each d2, any group Gd containing a d subgroup Sd of index 8 and exponent at most 2 such that Ud is contained 2 in a subgroup of Sd isomorphic to Z4 and such that, for d>2, SdÂUd contains a subgroup of index 4 and exponent at most 2d&1.

(ii) When j is even. For each d1, any group Gd containing a sub- r+jÂ2 d group Sd of index 2 and exponent at most 2 such that, for d>1, SdÂUd contains a subgroup of index 2r and exponent at most 2d&1 and such that, for j>0 and d>1, rank(Sd ÂUd)2r+jÂ2.

(iii) When j0, rank(Sd ÂUd)2r+j.

(iv) When j=r. For each d2, any group Gd containing a subgroup 2r d Sd of index 2 and exponent at most 2 such that Ud is contained in a sub- r group of Sd isomorphic to Z4 and such that Sd ÂUd contains a subgroup of index 2r and exponent at most 2d&1.

(v) When j>r. For each d1, any group Gd containing a subgroup r d+1 Sd of index 2 and exponent at most 2 such that Ud is contained in a sub- r group of Sd isomorphic to Z4 and such that Sd ÂUd contains a subgroup of index 2 j&r and exponent at most 2d. Proof. We apply Theorem 2.2 to certain of the families of BSs constructed in Corollaries 7.7, 7.9 and 7.10, putting p=2 and ignoring the initial BSs. For (i), use the BSs from Corollary 7.10 with c=d. For (ii), use the BSs from Corollary 7.7 with c=d and set 2i=j. For (iii), use the BSs from Corollary 7.9 with c=d and set i=j+r. For (iv), use the BSs from Corollary 7.9 with c=d&1 and d3, set i=j&r=0 and then replace d by d+1 throughout. For (v), use the BSs from Corollary 7.9 with c=d and d2, set i=j&r and then replace d by d+1 throughout. K The case j=r is dealt with separately from the case j>r in Corollary 8.4 because, when i=0, the set of BSs obtained from Corollary 7.9 with

File: 582A 279654 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3061 Signs: 2410 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 67 c=d&1 strictly contains the set of BSs obtained with c=d, by applying Lemma 2.1 with s=2r (whereas when i>0 this is not the case). Let P(i) be the number of partitions of the positive integer i. Then we 2r can take j=r and Sd$Z 2 d in Corollary 8.4(iv) to show that for each d2 there exists a (2(2d+1)r,2r,2(2d+1)r,22dr) semi-regular RDS in P(2r) non- r isomorphic groups Gd of rank 2r relative to any subgroup Z2 , including 2r 2r&1 Gd=Z2 d+1 and Gd=Z2 d+2r_Z2d . This shows that although the group rank must be at least 2r in order to use the underlying construction of Theorem 4.3, it need not grow any larger. To illustrate in detail how Corollary 8.4 improves on previous results, take r=2 and d=3 and consider which abelian groups G contain a (212+j, 12+j 10+j 2 4, 2 ,2 ) semi-regular RDS relative to a subgroup U$Z2 . We shall refer to the results of Table I, using (V) to indicate the condition that U is 2 contained in a subgroup of G isomorphic to Z4 . When j=0, the RDS parameters are (212,4,212,210) and G has order 214. Previously it was known that G could be any group of rank at least 8 using (B), any group of rank at least 7 such that (V) is satisfied using (C), 4 or the group U_Z8 of rank 6 using (E) with :=2, d=3. We can now use part (i) of Corollary 8.4 to include five groups G of rank 4, namely 2 3 3 2 2 Z64_Z8_Z4 , Z32_Z16_Z8_Z4 , Z32_Z8 , Z16_Z4 and Z16_Z18 for any 2 U$Z2 , as well as many new groups of rank 5 and 6 such that (V)is satisfied. Part (ii) of Corollary 8.4 is less powerful than part (i) in terms of large exponent, but it does not require the condition (V)onG. Parts (i) and (ii) together show that G can be any group of exponent at most 16 and 2 3 U any subgroup, except when G=Z2_Z16 and U intersects one or both of the first two direct factors of G in a nonidentity element. When j=1, the RDS parameters are (213,4,213,211) and G has order 215. Assume that G and U mentioned in this paragraph satisfy (V), which is necessary by Lemma 7.4. Previously it was known that G could be any group of rank at least 10 and exponent at most 4 using (F), or any group 2 of rank at least 8 having the form Z4_Z2_G$ (where U is contained in the first two direct factors) using the RDS product construction on (B) with :=5 and (A) with :=3. We can now use part (iii) of Corollary 8.4 3 2 2 to include six groups of rank 5, namely Z32 _Z8 _Z2 , Z16 _Z8 _Z2 , 3 5 2 2 2 2 Z16_Z8_Z4 and Z8 for any U, and Z32_Z8_Z4 and Z16_Z8_Z4 for any U which is not contained in the last two direct factors, as well as many new groups of rank 6 and higher. Part (iii) also shows that G can be any group of exponent at most 8 and U any subgroup. When j=2, the RDS parameters are (214,4,214,212) and G has order 216. Previously it was known that G could be any group of rank at least 9 using (B), any group of rank at least 8 such that (V) is satisfied using (C), 2 4 or the group Z4_Z8 of rank 6 where U is contained in the first two direct factors using the RDS product construction on (A) with :=2 and (E) with

File: 582A 279655 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3618 Signs: 3026 . Length: 45 pic 0 pts, 190 mm 68 DAVIS AND JEDWAB

:=2, d=3. We can now use part (iv) of Corollary 8.4 to include five groups of rank 4 (five being the number of partitions of 2r=4), namely 3 2 2 2 2 4 Z128_Z8 , Z64_Z16_Z8 , Z32_Z18 , Z32_Z16_Z8 and Z16 for any U,as well as many new groups of rank 5, 6 and 7 such that (V) is satisfied. We can also use part (ii) of Corollary 8.4 to include examples not satisfying (V) 2 for which G has low rank, for example G=Z64_Z2_Z8_Z4_Z2 where U is contained in the first two direct factors of G. When j=3, the RDS parameters are (215,4,215,213) and G has order 217. Assume that all G and U mentioned in this paragraph satisfy (V), which is necessary by Lemma 7.4. Previously it was known that G could be any group of rank at least 9 using (D) with :=3, d=2, or the group 2 4 Z4_Z2_Z8 of rank 7 (where U is contained in the first two direct factors) using the RDS product construction on (A) with :=3 and (E) with :=2, d=3. We can now use part (v) of Corollary 8.4 to include three groups of 3 2 2 rank 4, namely Z32_Z16 for any U, and Z64_Z16_Z8 and Z32_Z16_Z8 for any U which does not intersect the last direct factor in a nonidentity element, as well as many new groups of rank 5 to 8. Part (v) also shows that G can be any group of exponent at most 16 and U any subgroup. Finally we apply Theorem 2.2 to the Corollaries of Section 7 for p odd. This also gives substantial improvements over previous results.

Corollary 8.5. Let p be an odd prime. There exists a ( p2dr+j, pr, 2dr+j (2d&1) r+j p , p ) semi-regular RDS in the following groups Gd of order (2d+1) r+j r p relative to any subgroup Ud$Zp , where j and r satisfy 0j<2r:

(i) When j is even. For each d1, any group Gd containing a sub- r+jÂ2 d group Sd of index p and exponent at most p such that, for d>1, SdÂUd contains a subgroup of index pr and exponent at most pd&1 and such that, for j>0 and d>1, rank(Sd ÂUd)2r+jÂ2.

(ii) When j2, SdÂUd contains a subgroup of index pr and exponent at most pd&2 and such that, for d>2, rank(Sd ÂUd )4r+j.

(iii) When jr. For each d1, any group Gd containing a subgroup r d Sd of index p and exponent at most p such that, for d>1, SdÂUd contains a subgroup of index pr and exponent at most pd&1 and such that, for d>1, rank(Sd ÂUd)2r+j. Proof. The proof is similar to that of Corollary 8.4, using Corollaries 7.7 and 7.8. K

2 In particular, we can take j=0 and Sd$Zp d in Corollary 8.4(ii) and Corollary 8.5(i) to show that for each d1 and for any prime p there exists a(p2dr, pr, p2dr, p(2d&1)r) semi-regular RDS in P(r) nonisomorphic groups

File: 582A 279656 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3341 Signs: 2629 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 69

r r r Gd of rank 2r relative to any subgroup Zp , including Gd=Zp d+1_Zpd and 2r&1 Gd=Zp d+r_Zpd .(P(r) represents the number of partitions of r.)

9. OPEN PROBLEMS AND NONABELIAN CONSTRUCTIONS

We briefly note here some developments which occurred after submission of the original manuscript. 1. Chen [5, 6] constructed a new family of difference sets with parameters

q2d+2&1 2(q2d+2&1) (v, k, *, n)= 4q2d+2 , q2d+1 +1 , \ \ q2&1 + \ q+1 + q2d+1+1 q2d+1(q&1) , q4d+2 (13) \ q+1 + + for integer d0 and q a power of 2, 3 or p2, where p is an odd prime. With d=0 the parameters (13) correspond to the Hadamard parameters with N=q; with q=2 the parameters (13) correspond to the new family of parameters (4); and with q=3 the parameters (13) correspond to the Spence parameters with d replaced by 2d+1. Expressed in terms of the definitions of this paper, the difference sets constructed by Chen arise from a (q2d+1((q&1)Â2), q2d+1,4((q2d+2&1)Â (q2&1)), +) covering EBS on the elementary abelian group of order q2d+2 for q=3r or p2r, and from a (q2d+1(q&1), q2d+1,2((q2d+2&1)Â(q2&1)), +) covering EBS on the elementary abelian group of order 2q2d+2 for q=2r. The authors [14] have shown that these covering EBSs can be recursively constructed within the unifying framework of this paper by modifying Theorems 3.2 and 4.3 to deal with (a, m, t) BSs having m{- at. Such BSs always give rise to semi-regular divisible difference sets. 2. Jungickel and Schmidt [30] proposed that difference sets whose parameters (v, k, *, n) satisfy gcd(v, n)>1 should be considered as a special class. Indeed the five known parameter families satisfying this condition (namely Hadamard, McFarland, Spence, and parameter families (4) and (13)) are also the five families which have been shown to be amenable to the methods of this paper. 3. Schmidt [50] reduced the exponent bound in Corollary 5.5 from 8 to 4. Combined with Corollary 5.3(iv) this gives a necessary and sufficient condition for the existence of difference sets with parameters (4), provided that 2 is self-conjugate modulo the group exponent, with the possible

File: 582A 279657 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 2787 Signs: 2095 . Length: 45 pic 0 pts, 190 mm 70 DAVIS AND JEDWAB

3 exception of the group Z4_Z5 . Schmidt [50] also obtained exponent bounds for difference sets with parameters (13). 4. Davis, Jedwab and Mowbray [15] constructed new families of semi-regular RDSs in groups whose order is not a prime power, and applied the recursive construction of Theorem 7.11 followed by Theorem 2.2 to obtain further such families. The forbidden subgroup in these constructions has order 2r or 3, whereas previously the only known examples were those of Corollary 8.1 in which the forbidden subgroup has order 2.

In the remainder of this section, we list some open problems which are suggested by the techniques and results of this paper. We then discuss two possible approaches to generalising the definitions and constructions to deal with nonabelian groups. It appears from our results that the objects we have called BSs and covering EBSs are fundamental to the construction of difference sets and semi-regular RDSs. We believe that future research could usefully consider the following questions (all groups are still implicitly abelian):

1. Can the unifying framework of this paper be extended to encom- pass all known parameter families of difference sets, including projective geometries, the Paley-Hadamard family and the twin prime power family? 2. Can we find suitable BSs and covering EBSs for use in Theorem 3.2 (modified as in [14] if necessary) to construct difference sets via Theorem 2.4 whose parameters do not belong to any currently known family? 3. The construction of Hadamard difference sets in Section 6 relies on the existence of a (m((m&1)Â2), m, 4, +) covering EBS on a group of odd order m2. Can we find any examples apart from those of Theorem 6.6 and their compositions under Theorem 6.5? 4. The construction of Hadamard difference sets in Section 6 for which n=N 2 is not a prime power depends on Theorem 6.5. Is there an analogous composition theorem for McFarland difference sets or for difference sets with parameters (13)? 5. A construction for McFarland difference sets with q=2r in Sec- tion 5 relies on the existence of a (2qd, qd, qdÂ2) BS on a group G of order 2qd+1 relative to a subgroup of order q. Can we find any examples for q>4 (d+1) r+1 (d+1) r&1 apart from on G=Z 2 and Z4_Z2 ? In particular, can we find a (16, 8, 4) BS on a group of order 128 and exponent 4 (other than 5 Z4_Z2) relative to a subgroup of order 8, or a (32, 16, 8) BS on a group 7 of order 512 and exponent 4 (other than Z4_Z2) relative to a subgroup of order 16?

File: 582A 279658 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3006 Signs: 2500 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 71

3 6. Is there a (320, 88, 24, 64)-difference set in Z4_Z5 , the single exceptional group of Corollary 5.3(iv)? Theorem 3.2 would establish exist- 3 2 ence for this group if a (16, 8, 4) BS on Z4 relative to either Z4 or Z2 could be found. 7. (Due to Jungnickel and Schmidt [30]) Most exponent bounds for difference sets in the class gcd(v, n)>1 depend on a self-conjugacy condition which does not necessarily hold. Can we find an example for which such an exponent bound is exceeded? 8. Examples of semi-regular RDSs are known for which the forbidden subgroup is not elementary abelian (see the comment following Theorem 8.3). Can Theorem 4.3 be modified to construct BSs relative to a subgroup which is not elementary abelian to bring these RDSs within the framework of this paper? 9. The construction of families of BSs in Section 7 and semi-regular RDSs in Section 8 relies on the existence of initial (a, - at, t) BSs. Can we find any examples apart from those of Theorem 7.6 and those mentioned after Corollary 7.10? In particular, is the (8, 4, 2) BS of Theorem 7.6(iv) r the case r=2 of an infinite family of BSs relative to a subgroup U$Z 2 ? 2d+1 2d+1 2d 2 10. Is there a ( p , p, p , p ) semi-regular RDS in Zp d+1 or U_Zp d+1_Zpd relative to a subgroup U of order p, these being the two exceptional cases of Corollary 8.2(ii)? 11. We have argued in Section 8 that it is more appropriate to consider exponent bounds for a BS than for a semi-regular RDS. What can be said about a ( pd+w, pd, pd&w) BS relative to a subgroup of order pr ? We have seen in Section 1 that the existence pattern for difference sets in nonabelian groups is fundamentally different from that in abelian groups. We now consider how to modify the methods of this paper to deal with nonabelian groups, dropping the implicit assumption that all groups are abelian. Our first approach is based on techniques due to Dillon [20], and generalises Theorems 2.2 and 2.4:

Theorem 9.1. Let G be an abelian group. (i) Suppose there exists a (a, - at, t) BS on G relative to a subgroup U of order u, where at>1. Then there exists a (at, u, at, atÂu) semi-regular RDS in G$ relative to U, where G$ is any (possibly nonabelian) group containing a central subgroup of index t isomorphic to G. (ii) Suppose there exists a (a, m, h,\)covering EBS on G. Then there exists a (h |G|, ah\m, ah\m&m2, m2)-difference set in any (possibly nonabelian) group G$ containing a central subgroup of index h isomorphic to G.

File: 582A 279659 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3168 Signs: 2470 . Length: 45 pic 0 pts, 190 mm 72 DAVIS AND JEDWAB

Proof. We give the proof of (ii), the proof of (i) being similar. Let k1=1G$, k2, ..., kh # G$ be coset representatives of G in G$. Let [Bi ] be a (a, m, h, \) covering EBS on G, where the building block containing a\m elements is B1 , and form the subset D=i ki Bi of G$. If G$ is abelian then we already know Theorem 2.4 and the proof of Lemma 2.3 that D is a difference set in G$ with the stated parameters. By the characterisation of difference sets in the group ring Z[G$] this is equivalent to DD(&1)= 2 2 m 1G$+*G$, where *=ah\m&m =(aÂ|G|)(ah\2m) (using the relationship between covering EBS parameters given after Theorem 2.4 for the last equality). We can therefore obtain the result by showing that the same equation for DD(&1) holds when G$ is nonabelian as when G is abelian. By the definition of D we can write DD(&1) as the sum of two group ring elements:

(&1) (&1) &1 (&1) &1 DD =: kiBiBi ki + : ki Bi Bj kj . i i{j Since G is a central subgroup of G$, the first group ring element (&1) &1 (&1) i ki Bi B i ki is equal to i Bi Bi whether G$ is nonabelian or abelian, as required. It remains to consider the second group ring element S= (&1) &1 i{j ki Bi Bj kj , taking i{j. For all nonprincipal characters / of G we (&1) have /(Bi Bj )=/(Bi) /(Bj)=0, where the last equality uses the defini- (&1) tion of covering EBS. Therefore Bi Bj =cij G for some integer cij , and by a counting argument cij=|Bi||Bj|Â|G|. The definition of covering EBS 2 gives cij=a(a\m)Â|G| when i or j is 1 and cij=a Â|G| otherwise. Since (&1) G is central and therefore normal in G$, substitution for Bi Bj in S &1 gives the sum i{j cij ki kj G. The terms of this sum having j=1 are (aÂ|G|)(a\m) u{1 kuG. Likewise the terms of the sum having i=1 are &1 &1 j{1 c1j1G$kj G=(aÂ|G|)(a\m) u{1 kuG, since j{1 kj G=u{1 kuG. 2 &1 The remaining terms of the sum are (a Â|G|) i{j | i, j{1 kikj G. Regarding &1 &1 ki kj G as a coset of G, we can write ki kj G as the product of cosets &1 (ki G)(kjG) . Since the cosets [kuG | u{1] form a (h, h&1, h&2, 1)- difference set in G$ÂG (the complement of the trivial (h, 1, 0, 1)-difference 2 set [k1G] in G$ÂG), the remaining terms of the sum are (a Â|G|) 2 (h&2) u{1 kuG=(a Â|G|)(h&2)(G$&G). Therefore S=(aÂ|G|)(ah\2m) (G$&G)=*(G$&G), which holds whether G$ is nonabelian or abelian. This completes the proof. K The constructions of difference sets and semi-regular RDSs in abelian groups given in this paper can be extended to numerous nonabelian groups using Theorem 9.1 (including the special case when G$ can be written in the form G_K for some group K.) In the following discussion we shall concentrate on the construction of difference sets in nonabelian groups using

File: 582A 279660 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3377 Signs: 2620 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 73

Theorem 9.1(ii), but equally we can obtain semi-regular RDSs in many nonabelian groups by applying Theorem 9.1(i) to the BS families constructed in Corollaries 7.3 and 7.77.10. We begin by using the covering EBSs constructed in Theorem 5.2 to obtain difference sets in nonabelian groups with parameters from the McFarland family, Spence family or the new family (4). Applying Theorem 9.1(ii) to these covering EBSs shows that Corollary 5.3 remains true for nonabelian groups provided each occurrence of ``subgroup'' is replaced by ``central subgroup''. For example, Corollary 5.3(i) becomes: for each d0, there exists a McFarland difference set with q=pr in any (possibly nonabelian) group of order qd+1((qd+1&1)Â(q&1)+1) containing a central (d+1)r subgroup isomorphic to Zp , where p is prime and r1. This result was given by Dillon [20] and was used in Section 2 to introduce building blocks. We could similarly apply Theorem 9.1(ii) to the covering EBSs of Corollary 6.7(iii) to obtain difference sets in nonabelian groups with Hadamard parameters. But we can obtain more general results than this for Hadamard difference sets by taking advantage of a family of BSs constructed in Section 7:

Theorem 9.2. Let M be either the trivial group or the group 2 4 2 > Z : i_> Z , where : 1 and where p is an odd prime, and let |M|=m . i 3 j pj i j Then the following exist:

d+c&1 2 d d&c+2 (i) A (2 m ,2m,2 ,&)covering EBS on Gd, c_M, where d+c d and c satisfy 1cd and Gd, c is any abelian group of order 2 and exponent at most 2c. (ii) A Hadamard difference set with N=2dm in any ( possibly non- 2d+2 2 abelian) group Gd_M of order 2 m , where either d=0 or Gd contains a central subgroup of order 2d+c and exponent at most 2c such that d and c satisfy 1cd.

Proof. For (i), the proof is by induction on d. The case c=d has already been proved in Corollary 6.7(iii), so the case d=1 is true and we can take cd&1. Assume the case d&1 to be true. By Corollary 7.3(i) d+c&1 2 d d&c+1 there is a (2 m ,2m,2 )BSonGd,c_Mrelative to any subgroup Ud, c of order 2, and by the inductive hypothesis there is a d+c&2 2 d&1 d&c+1 (2 m ,2 m,2 , &) covering EBS on (Gd, c ÂUd, c)_M (since cd&1). The case d then follows from Theorem 3.2. For (ii), the case d=0 has already been proved in Corollary 6.7(iv). All other cases are obtained by applying Theorem 9.1(ii) to the covering EBSs of (i). K

File: 582A 279661 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 2865 Signs: 2380 . Length: 45 pic 0 pts, 190 mm 74 DAVIS AND JEDWAB

Theorem 9.2(ii) establishes the existence of Hadamard difference sets in large classes of nonabelian groups. Each value of c in Theorem 9.2(ii) produces examples which are not found at any other value of c. For larger values of c the required order of the central subgroup becomes larger but the group is allowed to have smaller rank. The case c=d of Theorem 9.2(i), for which the number of building blocks is 4, was given in Corollary 6.7(iii) and used to construct Hadamard difference sets in abelian groups in Corollary 6.7(iv). The cases c

File: 582A 279662 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3496 Signs: 3030 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 75

Hadamard difference set, because Davis and Smith [17] have constructed d 2d+3 a Hadamard difference set with N=2 in the group Gd=(x, y | x = 2d&1 &1 2 d+2+1 y =1, yxy =x ) for each d2, and Gd does not contain a normal abelian subgroup of order 2d+c and exponent at most 2c for any c satisfying 1cd. Furthermore the existence of a Hadamard difference set in Gd cannot rely on the existence of a covering EBS on an abelian group contained as a normal subgroup in Gd because Theorem 9.1(ii) would then give a Hadamard difference set in the abelian group (x, y | x2d+3= y2d&1=1, yx=xy) of order 22d+2 and exponent 2d+3, which is ruled out by Turyn's exponent bound [55]. This provides the motivation for our second (more speculative) approach to nonabelian groups, namely to generalise the definition of building block, BS and covering EBS to allow the group G in Theorem 9.1 to be nonabelian. Liebler [34] has promoted the use of representation theory to study difference sets in nonabelian groups, as a natural generalisation of the use of character theory for abelian groups. A representation , of a group G is a homomorphism from G to the multiplicative group of s_s matrices, where the degree s of the representation is determined by G. Lemma 1.1(i) generalises to: the k-element subset D of a group G of order v is a (v, k, *, n)-difference set in G if and only if ,(D) ,(D)$=-nIs for every nontrivial irreducible representation , of G, where ,(D)$ is the conjugate transpose of ,(D) and Is is the s_s identity matrix. We might define a building block B in a group G with modulus m to be a subset of G such that for all nontrivial irreducible representations ,, the representation sum

,(B)=g # B ,(g) is either 0 or satisfies ,(B) ,(B)$=mIs . Then a (a, m, t) building set on G relative to U would be a collection of t building blocks in G with modulus m, each containing a elements, such that for every nontrivial irreducible representation , of G exactly one building block has nonzero representation sum if , is nontrivial on U and no building block has nonzero representation sum if , is trivial on U. We could similarly extend the definition of EBS. Although we believe this approach could be fruitful, it will not allow us to replace ``central'' by ``normal'' in Theorem 9.1(i), even when G is still assumed to be abelian. For example, by Theorem 4.2 2 there is a (2, 2, 2) BS on Z2 relative to Z2 , but there is no (4, 2, 4, 2) RDS in D8 (the dihedral group of order 8) relative to the central subgroup of order 2 even though D8 contains a normal subgroup of index 2 isomorphic 2 to Z2 . The difficulty appears to arise because the restriction of an irreducible representation to a normal subgroup is not necessarily irreducible. The following construction of Meisner [44] for Hadamard difference sets (which generalises the result of applying Theorem 2.4 to the case h=t=1 of Theorem 3.2) may be of importance in formulating a general nonabelian approach to the construction of difference sets and semi-regular RDSs involving representation theory.

File: 582A 279663 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 3503 Signs: 3077 . Length: 45 pic 0 pts, 190 mm 76 DAVIS AND JEDWAB

Theorem 9.3. Suppose that there exists a (4N2,2,4N2,2N2)semi- regular RDS in a group H of order 8N 2 relative to a central subgroup (x) of order 2. Suppose also that there exists a Hadamard difference set with parameter N in HÂ(x). Then there exists a Hadamard difference set with parameter 2N in any group G$ containing H as a subgroup of index 2 for which x is a central element of G$. Meisner [4446] has shown that Theorem 9.3 (together with a partial generalisation of Theorem 4.3) can be used to construct Hadamard difference sets in certain nonabelian groups containing a normal abelian subgroup M, as used in Theorem 9.2(ii), for which M is not a central subgroup. These fall outside the scope of Theorem 9.2(ii).

ACKNOWLEDGMENT

We are grateful to Miranda Mowbray for her careful reading of the manuscript and for many helpful comments.

REFERENCES

1. S. Alquaddoomi and R. A. Scholtz, On the nonexistence of Barker arrays and related mat- ters, IEEE Trans. Inform. Theory 35 (1989), 10481057. 2. K. T. Arasu, J. A. Davis, J. Jedwab, and S. K. Sehgal, New constructions of Menon difference sets, J. Combin. Theory Ser. A 64 (1993), 329336. 3. K. T. Arasu and S. K. Sehgal, Some new difference sets, J. Combin. Theory Ser. A 69 (1995), 170172. 4. T. Beth, D. Jungnickel, and H. Lenz, ``Design Theory,'' Cambridge University Press, Cambridge, 1986. 5. Y. Q. Chen, On the existence of abelian Hadamard difference sets and a new family of difference sets, Finite Fields Appl., to appear. 6. Y. Q. Chen, A construction of difference sets, Designs, Codes, Cryptogr., to appear. 7. Y. Q. Chen, D. K. Ray-Chaudhuri, and Q. Xiang, Constructions of partial difference sets and relative difference sets using Galois rings II, J. Combin. Theory Ser. A 76 (1996), 179196. 8. J. A. Davis, A note on products of relative difference sets, Designs, Codes, Cryptogr. 1 (1991), 117119. 9. J. A. Davis, A result on Dillon's conjecture in difference sets, J. Combin. Theory Ser. A 57 (1991), 238242. 10. J. A. Davis, Construction of relative difference sets in p-groups, Discrete Math. 103 (1992), 715. 11. J. A. Davis, An exponent bound for relative difference sets in p-groups, Ars Combin. 34 (1992), 318320. 12. J. A. Davis and J. Jedwab, Recursive construction for difference sets, Bull. Inst. Combin. Appl. 13 (1995), 128, research announcement.

File: 582A 279664 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 5288 Signs: 2343 . Length: 45 pic 0 pts, 190 mm DIFFERENCE SETS CONSTRUCTION 77

13. J. A. Davis and J. Jedwab, A survey of Hadamard difference sets, in ``Groups, Difference Sets and the Monster'' (K. T. Arasu et al., Eds.), pp. 145156, de Gruyter, BerlinÂNew York, 1996. 14. J. A. Davis and J. Jedwab, Some recent developments in difference sets, in ``Combinatorial Designs and Their Applications'' (K. Quinn et al., Eds.), Pitman Research Notes in Mathematics, AddisonWesleyLongman, London, to appear. 15. J. A. Davis, J. Jedwab, and M. Mowbray, New families of semi-regular relative difference sets, Designs, Codes and Cryptogr., to appear. 16. J. A. Davis and S. K. Sehgal, Using the Simplex code to construct relative difference sets in 2-groups, Designs, Codes, and Cryptogr. 11 (1997), 267277. 17. J. A. Davis and K. W. Smith, A construction of difference sets in high exponent 2-groups using representation theory, J. Algebraic Combin. 3 (1994), 137151. 18. J. F. Dillon, A survey of difference sets in 2-groups, presented at ``Marshall Hall Memorial Conference, Vermont, 1990.'' 19. J. F. Dillon, ``Elementary Hadamard Difference Sets,'' Ph.D. thesis, University of Maryland, 1974. 20. J. F. Dillon, Variations on a scheme of McFarland for noncyclic difference sets, J. Combin. Theory Ser. A 40 (1985), 921. 21. J. E. H. Elliott and A. T. Butson, Relative difference sets, Illinois J. Math. 10 (1966), 517531. 22. M. van Eupen and V. D. Tonchev, Linear codes and the existence of a reversible 4 Hadamard difference set in Z2_Z2_Z5 , J. Combin. Theory Ser. A 79 (1997), 161167. 23. E. E. Fenimore and T. M. Cannon, Coded aperture imaging with uniformly redundant arrays, Appl. Optics 17 (1978), 337347. 24. M. J. Ganley, On a paper of Dembowski and Ostrom, Arch. Math. 27 (1976), 9398. 25. J. E. Hershey and R. Yarlagadda, Two-dimensional synchronisation, Electron. Lett. 19 (1983), 801803. 26. A. J. Hoffman, Cyclic affine planes, Canad. J. Math. 4 (1952), 295301. 27. J. Jedwab, Generalized perfect arrays and Menon difference sets, Designs, Codes and Cryptogr. 2 (1992), 1968. 28. D. Jungnickel, On automorphism groups of divisible designs, Canad. J. Math. 34 (1982), 257297. 29. D. Jungnickel, Difference sets, in ``Contemporary Design Theory: A Collection of Surveys'' (J. H. Dinitz and D. R. Stinson, Eds.), pp. 241324, Wiley, New York, 1992. 30. D. Jungnickel and B. Schmidt, Difference sets: An update, in ``Geometry, Combinatorial Designs and Related Structures'' (J. W. P. Hirschfeld, S. S. Magliveras, and M. J. de Resmini, Eds.), to appear. 31. R. G. Kraemer, Proof of a conjecture on Hadamard 2-groups, J. Combin. Theory Ser. A 63 (1993), 110. 32. E. S. Lander, ``Symmetric Designs: An Algebraic Approach,'' London Math. Society Lec- ture Notes, Vol. 74, Cambridge University Press, Cambridge, 1983. 33. K. H. Leung and S. L. Ma, Constructions of partial difference sets and relative difference sets on p-groups, Bull. London Math. Soc. 22 (1990), 533539. 34. R. A. Liebler, The inversion formula, J. Combin. Math. Combin. Comput. 13 (1993), 143160. 35. R. A. Liebler and K. W. Smith, On difference sets in certain 2-groups, in ``Coding Theory, Design Theory, Group Theory'' (D. Jungnickel and S. A. Vanstone, Eds.), pp. 195212, Wiley, New York, 1993. 36. S. L. Ma and A. Pott, Relative difference sets, planar functions and generalized Hadamard matrices, J. Algebra 175 (1995), 505525.

File: 582A 279665 . By:DS . Date:15:09:97 . Time:09:17 LOP8M. V8.0. Page 01:01 Codes: 8728 Signs: 3322 . Length: 45 pic 0 pts, 190 mm 78 DAVIS AND JEDWAB

37. S. L. Ma and B. Schmidt, On ( pa, p, pa, pa&1)-relative difference sets, Designs, Codes and Cryptogr. 6 (1995), 5771. 38. S. L. Ma and B. Schmidt, ``A Sharp Exponent Bound for McFarland Difference Sets with p=2,'' preprint, National University of Singapore, 1995. 39. S. L. Ma and B. Schmidt, The structure of the abelian groups containing McFarland difference sets, J. Combin. Theory Ser. A 70 (1995), 313322. 40. S. L. Ma and B. Schmidt, Difference sets corresponding to a class of symmetric designs, Designs, Codes and Cryptogr. 10 (1997), 223236. 41. S. J. Martin, M. A. Butler, and C. E. Land, Ferroelectric optical image comparator using PLZT thin films, Electron. Lett. 24 (1988), 14861487. 42. R. L. McFarland, A family of difference sets in non-cyclic groups, J. Combin. Theory Ser. A 15 (1973), 110. 43. D. B. Meisner, ``Menon Designs and Related Difference Sets,'' Ph.D. thesis, University of London, 1991. 44. D. B. Meisner, Families of Menon difference sets, Ann. Discrete Math. 52 (1992), 365380. 45. D. B. Meisner, New classes of groups containing Menon difference sets, Designs, Codes and Cryptogr. 8 (1996), 319325. 46. D. B. Meisner, A difference set construction of Turyn adapted to semi-direct products, in ``Groups, Difference Sets and the Monster'' (K. T. Arasu et al., Eds.), pp. 169174, de Gruyter, BerlinÂNew York, 1996. 47. A. Pott, On the structure of abelian groups admitting divisible difference sets, J. Combin. Theory Ser. A 65 (1994), 202213. 48. A. Pott, A survey on relative difference sets, in ``Groups, Difference Sets and the Monster'' (K. T. Arasu et al., Eds.), pp. 195232, de Gruyter, BerlinÂNew York, 1996. 49. B. Schmidt, On ( pa, pb, pa, pa&b)-relative difference sets, J. Algebraic Combin. 6 (1997), 279297. 50. B. Schmidt, Nonexistence results on Chen and DavisJedwab difference sets, J. Algebra, to appear. 51. J. Seberry and M. Yamada, Hadamard matrices, sequences, and block designs, in ``Con- temporary Design Theory: A Collection of Surveys'' (J. H. Dinitz and D. R. Stinson, Eds.), pp. 431560, Wiley, New York, 1992. 52. G. K. Skinner, X-ray imaging with coded masks, Scientific American 259 (Aug. 1988), 6671. 53. K. W. Smith, Non-abelian Hadamard difference sets, J. Combin. Theory Ser. A 70 (1995), 144156. 54. E. Spence, A family of difference sets, J. Combin. Theory Ser. A 22 (1977), 103106. 55. R. J. Turyn, Character sums and difference sets, Pacific J. Math. 15 (1965), 319346. 56. R. J. Turyn, A special class of Williamson matrices and difference sets, J. Combin. Theory Ser. A 36 (1984), 111115. 57. R. M. Wilson and Q. Xiang, Constructions of Hadamard difference sets, J. Combin. Theory Ser. A 77 (1997), 148160. 58. M.-Y. Xia, Some infinite classes of special Williamson matrices and difference sets, J. Combin. Theory Ser. A 61 (1992), 230242. 59. Q. Xiang and Y. Q. Chen, On Xia's construction of Hadamard difference sets, Finite Fields Appl. 2 (1996), 8795.

File: 582A 279666 . By:DS . Date:15:09:97 . Time:09:18 LOP8M. V8.0. Page 01:01 Codes: 7793 Signs: 2930 . Length: 45 pic 0 pts, 190 mm